#363040

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

806 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=eg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^c=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, 270236);