#6576231

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1943 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)(13,14,19,23)(15,20,27,39)(16,21,30,17)(18,22,35,42)(24,25,37,43)(26,38,33,41)(28,40,44,31)(29,36,34,32);(6,9)(10,12)(13,15,19,27)(14,20,23,39)(16,28,30,44)(17,31,21,40)(18,33,35,26)(22,41,42,38)(24,34,37,29)(25,32,43,36);(2,4)(13,16,19,30)(14,21,23,17)(15,24,27,37)(18,34,35,29)(20,25,39,43)(22,32,42,36)(26,44,33,28)(31,41,40,38);(5,7,8,11)(6,10,9,12)(16,29)(17,32)(18,35)(21,36)(22,42)(24,28)(25,40)(26,33)(30,34)(31,43)(37,44)(38,41);(13,17,19,21)(14,16,23,30)(15,25,27,43)(18,36,35,32)(20,37,39,24)(22,34,42,29)(26,31,33,40)(28,41,44,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,e^c=ei,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, 6483427);