#6575210

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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

2245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6482406);