#6574563

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1924 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481759);