#6685620

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

1884 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)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,26,33)(16,20,28,38)(17,22,31,39)(23,32,41,43)(24,34,25,35)(27,36,42,44)(29,37,30,40);(6,9)(7,11)(13,15,18,26)(14,19,21,33)(16,27,28,42)(17,24,31,25)(20,36,38,44)(22,34,39,35)(23,30,41,29)(32,40,43,37);(1,2,3,4)(13,16)(14,20)(15,23)(17,30)(18,28)(19,32)(21,38)(22,40)(24,27)(25,42)(26,41)(29,31)(33,43)(34,36)(35,44)(37,39);(6,9)(10,12)(14,21)(16,28)(19,33)(22,39)(23,41)(27,42)(29,30)(34,35);(5,7,8,11)(6,10,9,12)(15,24)(17,31)(19,34)(22,39)(23,27)(25,26)(29,30)(32,36)(33,35)(37,40)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,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, 6592816);