#6574505

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1911 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)(5,6,8,10)(7,11,12,9);(5,7,8,12)(6,9,10,11)(13,14)(15,24)(16,27)(17,32)(18,35)(19,23)(20,26)(21,31)(22,37)(25,43)(28,44)(29,38)(30,41)(33,40)(34,39)(36,42);(2,4)(13,15,19,17)(14,20,23,39)(16,28,31,29)(18,34,37,26)(21,40,27,43)(22,32,35,24)(25,30,33,36)(38,41,44,42);(13,16,19,31)(14,21,23,27)(15,25,17,33)(18,36,37,30)(20,38,39,44)(22,42,35,41)(24,40,32,43)(26,28,34,29);(13,17,19,15)(14,20,23,39)(16,29,31,28)(18,26,37,34)(21,40,27,43)(22,32,35,24)(25,36,33,30)(38,41,44,42)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6481701);