#364344

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1462 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fi,d^2=egi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271540);