#6816481

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

1888 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6723677);