#6685621

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
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1851 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)(13,14,17,21)(15,23,26,19)(16,20,28,36)(18,22,30,38)(24,41,42,33)(25,39,31,34)(27,44,43,35)(29,37,32,40);(6,9)(7,11)(13,15,17,26)(14,19,21,23)(16,27,28,43)(18,31,30,25)(20,35,36,44)(22,39,38,34)(24,32,42,29)(33,40,41,37);(1,2,3,4)(13,16)(14,20)(15,24)(17,28)(18,32)(19,33)(21,36)(22,40)(23,41)(25,43)(26,42)(27,31)(29,30)(34,44)(35,39)(37,38);(6,9)(10,12)(16,28)(20,36)(24,42)(27,43)(29,32)(33,41)(35,44)(37,40);(5,7,8,11)(6,10,9,12)(15,25)(18,30)(19,34)(22,38)(23,39)(24,43)(26,31)(27,42)(29,32)(33,44)(35,41)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,e^2=fi,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, 6592817);