#6810623

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

1861 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)(15,20)(16,21)(17,19)(18,22)(23,25)(24,27)(26,28);(5,6)(7,12)(8,11)(9,10)(14,19)(15,23)(16,25)(18,27)(20,26)(21,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,18,23,17,21,27,28)(14,20,22,25,19,16,24,26);(5,8,9,7)(6,11,10,12)(14,19)(15,21)(22,24)(23,28);(13,16,18,26,17,20,27,25)(14,21,22,28,19,15,24,23)

Polycyclic presentation:

2^9|c^2=h,d^2=fi,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, 6717819);