#6808151

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=f,e^2=h,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, 6715347);