#95560

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

598 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=gh,e^2=i,f^2=hi,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,e^b=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2756);