#6516696

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,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, 6423892);