#6818714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1217 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fgh,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=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, 6725910);