#6808148

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

1961 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)(15,24)(16,18)(17,26)(19,23)(20,22)(21,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,19,25)(14,21,23,24)(16,26,27,22)(17,18,20,28);(2,4)(13,16,17,24,19,27,20,21)(14,18,26,15,23,28,22,25);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,22,23,26)(15,18,25,28)(16,24,27,21);(6,9)(10,12)(13,18,17,15,19,28,20,25)(14,16,26,24,23,27,22,21)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=f,e^2=hi,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, 6715344);