#6809154

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
prime radical	C₂×#3283	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,18)(15,20,25,33)(16,21,29,27)(17,22,32,31)(23,26,34,37)(24,30,36,43)(28,35,42,40)(38,39,41,44);(6,9)(7,11)(13,15,19,25)(14,20,18,33)(16,26,29,37)(17,30,32,43)(21,34,27,23)(22,36,31,24)(28,41,42,38)(35,44,40,39);(2,4)(13,16,14,21,19,29,18,27)(15,23,33,37,25,34,20,26)(17,28,22,35,32,42,31,40)(24,38,43,44,36,41,30,39);(6,9)(10,12)(13,14,19,18)(15,20,25,33)(16,27,29,21)(17,22,32,31)(23,37,34,26)(24,30,36,43)(28,40,42,35)(38,44,41,39);(5,7,8,11)(6,10,9,12)(13,17,14,22,19,32,18,31)(15,24,33,43,25,36,20,30)(16,28,21,35,29,42,27,40)(23,38,37,44,34,41,26,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=i,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, 6716350);