#6810617

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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

1892 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,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,d^c=di

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, 6717813);