#6816502

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 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)(15,23,24,17)(16,26,25,20)(19,22,27,28);(5,7,8,12)(6,9,10,11)(13,15)(14,17)(16,19)(18,24)(20,22)(21,23)(25,27)(26,28);(1,2,3,4)(6,10)(7,12)(13,16,27,24,18,25,19,15)(14,20,28,23,21,26,22,17);(6,10)(9,11)(14,21)(17,23)(20,26)(22,28);(13,17,19,26,18,23,27,20)(14,15,22,25,21,24,28,16)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,e^2=fhi,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, 6723698);