#6763547

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37701	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1635 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,22,27)(19,23,29,38)(20,32,35,43)(21,25,34,26)(24,40,42,36)(28,45,33,48)(30,39,31,44)(37,46,41,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,29)(18,23,27,38)(20,33,35,28)(21,31,34,30)(24,41,42,37)(25,44,26,39)(32,48,43,45)(36,46,40,47);(2,6)(3,7)(10,13)(14,16)(17,20,34,40,22,35,21,36)(18,24,25,43,27,42,26,32)(19,28,30,46,29,33,31,47)(23,37,44,48,38,41,39,45);(9,11,12,15)(10,14,13,16)(19,29)(20,35)(23,38)(24,42)(30,31)(32,43)(36,40)(39,44);(3,7)(5,8)(17,21,22,34)(18,25,27,26)(19,30,29,31)(20,36,35,40)(23,39,38,44)(24,43,42,32)(28,46,33,47)(37,45,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6670743);