#6462536

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6369732);