#6575803

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6482999);