#72232015

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	C₂×C₄	C₂³×D₈
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂²×C₄×D₈
prime radical	#7623249	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ C₈×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,d^2=f,e^2=gi,f^2=h,g^2=ij,i^2=j,b^a=bh,c^b=cj,e^a=eg,e^d=ej,g^a=gi,i^a=ij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232026, #72260420, #72260475, …
maximal normal subgroup of	#72232026, #72260420, #72260475, …

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