#6508606

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1507 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,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,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, 6415802);