#6682780

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

2245 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)(13,14,17,21)(15,23,27,19)(16,20,29,36)(18,22,31,38)(24,34,26,32)(25,35,28,33)(30,37,41,44)(39,43,40,42);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,28)(17,27)(18,24)(20,35)(21,23)(22,32)(25,29)(26,31)(30,40)(33,36)(34,38)(37,43)(39,41)(42,44);(1,2,3,4)(15,24)(18,31)(19,32)(22,38)(23,34)(25,39)(26,27)(28,40)(30,41)(33,42)(35,43)(37,44);(6,10)(9,11)(13,16)(14,20)(15,25)(17,29)(18,30)(19,33)(21,36)(22,37)(23,35)(24,39)(26,40)(27,28)(31,41)(32,42)(34,43)(38,44);(7,12)(9,11)(15,26)(16,29)(18,31)(19,34)(20,36)(22,38)(23,32)(24,27)(25,39)(28,40)(33,42)(35,43)

Polycyclic presentation:

2^9|a^2=i,c^2=g,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, 6589976);