#6809930

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2158 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)(5,6)(7,11)(8,10)(9,12)(13,14,18,21)(15,23,25,17)(16,27,26,20)(19,22,28,24);(5,7)(6,9)(8,12)(10,11)(13,15,18,25)(14,17,21,23)(16,28,26,19)(20,24,27,22);(2,4)(7,12)(9,11)(13,16,19,15,18,26,28,25)(14,20,22,17,21,27,24,23);(6,10)(9,11)(14,21)(17,23)(20,27)(22,24);(13,17,28,20,18,23,19,27)(14,15,24,16,21,25,22,26)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,e^2=fh,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, 6717126);