#72307234

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#17595
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂²×M₁₆	C₂⁴
prime radical	C₂×#3184	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #20331 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1583 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=ef,c^2=h,d^2=gj,e^2=i,g^2=j,i^2=j,b^a=be,c^b=cg,d^a=dij,d^b=dgi,d^c=dj,e^a=ei,e^b=eh,e^c=ej,f^a=fh,f^b=fh,f^e=fj,g^a=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307241, #72307340
maximal normal subgroup of	#72307241, #72307340

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.8);;
gap> SetPower(coll, 2, F.5*F.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.7*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);