#6519636

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500443
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	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1984 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:

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

Polycyclic presentation:

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

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, 6426832);