#160487

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cg,e^a=eg,e^b=eh,f^b=fgh

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, 67683);