#72321385

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#442
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂²×#509	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6,9,8,10,11,7,12)(13,14,18,24,20,25,21,37)(15,26,29,43,30,38,31,40)(16,32,33,42,34,22,28,39)(17,23,19,36,27,41,35,44);(2,4)(5,7,10,9)(6,8,11,12)(13,15,20,30)(14,22,25,32)(16,17,34,27)(18,31,21,29)(19,33,35,28)(23,40,41,43)(24,42,37,39)(26,36,38,44);(13,16)(14,22)(15,27)(17,30)(18,33)(19,31)(20,34)(21,28)(23,26)(24,39)(25,32)(29,35)(36,43)(37,42)(38,41)(40,44);(2,4)(5,8,9,11,10,12,7,6)(13,17,21,35,20,27,18,19)(14,23,37,44,25,41,24,36)(15,28,31,34,30,33,29,16)(22,38,42,43,32,26,39,40)

Polycyclic presentation:

2^10|a^2=e,b^2=h,d^2=efghij,e^2=h,f^2=i,g^2=i,i^2=j,b^a=bf,c^a=cj,c^b=cj,d^a=dfgij,d^b=dhij,d^c=dj,e^b=ej,f^a=fi,f^b=fij,f^d=fj,g^a=gi,g^b=gij,g^d=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321461, #72321521, #72321966
maximal normal subgroup of	#72321461, #72321521, #72321966
maximal quotient of	#72322592

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 4, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);