#6809401

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2357 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)(5,6)(7,10)(8,9)(11,12)(13,14)(15,16)(17,26)(18,28)(19,25)(20,23)(21,22)(24,27);(6,9)(7,11)(13,15,19,22)(14,21,25,16)(17,27,20,18)(23,24,26,28);(2,4)(13,16,20,24,19,21,17,28)(14,22,23,18,25,15,26,27);(6,9)(10,12)(13,17,19,20)(14,23,25,26)(15,27,22,18)(16,28,21,24);(5,7,8,11)(6,10,9,12)(13,18,17,22,19,27,20,15)(14,24,26,16,25,28,23,21)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6716597);