#6667407

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660434
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18,22,27)(19,23,29,36)(20,25,31,39)(21,24,33,26)(28,35,42,47)(30,34,43,37)(32,38,44,40)(41,45,48,46);(1,3,6,5)(2,7,4,8)(10,13)(14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,34)(25,35)(26,37)(27,36)(31,42)(32,41)(33,43)(38,45)(39,47)(40,46)(44,48);(1,4,6,2)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(18,24)(21,33)(23,34)(25,38)(26,27)(30,43)(32,44)(35,45)(36,37)(39,40)(41,48)(46,47);(2,4)(7,8)(10,13)(14,16)(17,20,22,31)(18,25,27,39)(19,28,29,42)(21,32,33,44)(23,35,36,47)(24,38,26,40)(30,41,43,48)(34,45,37,46);(1,5,6,3)(2,8,4,7)(9,11,12,15)(10,14,13,16)(19,29)(20,31)(23,36)(25,39)(30,43)(32,44)(34,37)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6574603);