#96982

order	512
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=eg,f^c=fg

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, 4178);