#6502218

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	#35564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1894 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)(17,18)(19,23)(20,28)(21,27)(22,25)(24,26)(29,30)(31,32);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,21)(18,23,25,27)(20,29,31,26)(24,28,30,32);(9,11)(10,13)(12,16)(14,15)(17,20,22,31)(18,24,25,30)(19,26,21,29)(23,32,27,28);(11,16)(13,15)(17,18)(19,27)(20,30)(21,23)(22,25)(24,31)(26,32)(28,29);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,19,22,21)(18,23,25,27)(20,26,31,29)(24,32,30,28)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6409414);