#6514209

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1833 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6421405);