#6466599

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=gh,d^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373795);