#6508574

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500443
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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)(19,24)(20,21)(22,29)(23,25)(26,30)(27,31)(28,32);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,27)(18,24,25,31)(20,26,29,32)(21,30,22,28);(9,11)(10,13)(12,16)(14,15)(17,20,21,18,23,29,22,25)(19,26,30,24,27,32,28,31);(11,16)(13,15)(19,27)(24,31)(26,32)(28,30);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,20,25,29)(19,28,27,30)(24,32,31,26)

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^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415770);