#6576198

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1980 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)(5,6,8,9)(7,10,11,12)(13,14,19,24)(15,20,28,37)(16,21,32,41)(17,22,27,36)(18,23,26,35)(25,34,30,39)(29,38,43,44)(31,40,33,42);(6,9)(10,12)(13,15,19,28)(14,20,24,37)(16,29,32,43)(17,26,27,18)(21,38,41,44)(22,35,36,23)(25,33,30,31)(34,42,39,40);(2,4)(13,16,19,32)(14,21,24,41)(15,25,28,30)(17,29,27,43)(18,33,26,31)(20,34,37,39)(22,38,36,44)(23,42,35,40);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,26)(16,30)(18,28)(19,27)(20,35)(21,39)(23,37)(24,36)(25,32)(29,31)(33,43)(34,41)(38,40)(42,44);(15,27)(17,28)(18,26)(20,36)(22,37)(23,35)(25,43)(29,30)(31,33)(34,44)(38,39)(40,42)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6483394);