#6574657

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1904 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)(14,20)(21,37)(22,38)(23,35)(24,31)(39,43)(40,44)(41,42);(5,7,8,12)(6,9,10,11)(13,14,18,20)(15,24,17,31)(16,22,29,38)(19,35,34,23)(21,33,37,26)(25,40,32,44)(27,39,28,43)(30,42,36,41);(2,4)(13,15,18,17)(14,21,20,37)(16,27,29,28)(19,33,34,26)(22,40,38,44)(23,31,35,24)(25,30,32,36)(39,41,43,42);(13,16,18,29)(14,22,20,38)(15,25,17,32)(19,36,34,30)(21,39,37,43)(23,42,35,41)(24,40,31,44)(26,27,33,28);(13,17,18,15)(14,21,20,37)(16,28,29,27)(19,26,34,33)(22,40,38,44)(23,31,35,24)(25,36,32,30)(39,41,43,42)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=i,e^2=i,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, 6481853);