#362440

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

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

1198 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=e,d^2=fi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 269636);