#127315

order	512
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	H(3,ℤ/4)
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₄×M₁₆	C₂³
socle	C₂	#7972
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=ghi,f^2=h,g^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,e^b=eh,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34511);