#72322121

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623096	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,9,6,10,4,8)(3,7,12,15,11,14,13,16)(17,18,20,26,22,24,33,27)(19,28,30,47,29,46,32,41)(21,35,34,25,38,45,37,44)(23,39,42,31,40,36,43,48);(1,3,4,13,6,11,5,12)(2,7,8,16,10,14,9,15)(17,19)(18,23,24,40)(20,32)(21,36)(22,29)(25,41,44,47)(26,43,27,42)(28,45,46,35)(30,33)(31,34)(37,48)(38,39);(19,29)(23,40)(28,46)(30,32)(31,48)(36,39)(41,47)(42,43);(3,11)(7,14)(12,13)(15,16)(18,24)(23,40)(25,44)(26,27)(28,46)(35,45)(41,47)(42,43);(1,4,6,5)(2,8,10,9)(3,12,11,13)(7,15,14,16)(18,25)(20,33)(21,37)(23,41)(24,44)(26,35)(27,45)(28,43)(30,32)(31,39)(34,38)(36,48)(40,47)(42,46)

Polycyclic presentation:

2^10|a^2=f,b^2=fhi,e^2=hj,f^2=h,g^2=i,i^2=j,b^a=bg,c^b=cj,d^a=dj,d^b=dhj,e^a=egij,e^b=ehj,f^b=fj,f^e=fj,g^a=gi,g^b=gi,g^e=gij,i^a=ij,i^b=ij,i^e=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322136, #72322467
maximal normal subgroup of	#72322136, #72322467
maximal quotient of	#72343866

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