#126594

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=g,e^2=h,h^2=i,b^a=bd,f^a=fg,c^b=ci,e^b=eg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33790);