#6815016

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2008 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)(6,9)(7,11)(13,14,18,21)(15,23,24,17)(16,25,27,20)(19,22,26,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,26)(18,24)(19,27)(20,28)(21,23)(22,25);(1,2,3,4)(13,16,26,15,18,27,19,24)(14,20,28,17,21,25,22,23);(5,7,8,11)(6,10,9,12)(14,21)(17,23)(20,25)(22,28);(6,9)(10,12)(13,17,19,20,18,23,26,25)(14,15,22,16,21,24,28,27)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=fi,e^2=h,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, 6722212);