#6682777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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

2242 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6589973);