#6466756

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1395 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,20)(21,26)(22,32)(23,28)(24,27)(25,30)(29,31);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,28)(21,32)(22,29)(23,30)(24,26)(27,31);(9,12,13,11)(10,14,15,16)(17,20,24,31,23,25,22,21)(18,19,27,29,28,30,32,26);(17,21,24,20,23,31,22,25)(18,26,27,19,28,29,32,30);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,27,28,32)(19,29,30,26)(20,21,25,31)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,e^2=f,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, 6373952);