#6667413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660408
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

1401 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=di,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, 6574609);