#6671513

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2167 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)(6,9)(7,11);(5,6,8,9)(7,10,11,12)(13,14)(15,24)(16,21)(17,30)(18,32)(19,23)(20,27)(22,33)(25,35)(26,41)(28,36)(29,38)(31,37)(34,42)(39,43)(40,44);(2,4)(13,15,19,28)(14,20,23,35)(16,27,18,25)(17,26,31,42)(21,36,32,24)(22,34,38,41)(29,43,33,40)(30,44,37,39);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,21,23,32)(15,25,28,27)(17,29,31,33)(20,24,35,36)(22,30,38,37)(26,40,42,43)(34,39,41,44);(6,9)(10,12)(13,17,19,31)(14,22,23,38)(15,26,28,42)(16,29,18,33)(20,34,35,41)(21,30,32,37)(24,39,36,44)(25,40,27,43)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6578709);