#6576831

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	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2677 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)(7,12)(8,10)(9,11);(5,7,9,8)(6,10,11,12)(13,14,18,22)(15,23,26,38)(16,27,30,36)(17,31,33,21)(19,32,35,24)(20,28,37,29)(25,34,41,43)(39,40,44,42);(2,4)(13,15,18,26)(14,19,22,35)(16,25,30,41)(17,32,33,24)(20,34,37,43)(21,38,31,23)(27,39,36,44)(28,42,29,40);(5,8,9,7)(6,10,11,12)(15,24)(16,28)(17,33)(19,23)(20,36)(21,31)(25,41)(26,32)(27,37)(29,30)(34,43)(35,38);(13,16,18,30)(14,20,22,37)(15,25,26,41)(17,28,33,29)(19,34,35,43)(21,36,31,27)(23,39,38,44)(24,40,32,42)

Polycyclic presentation:

2^9|b^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=dhi,h^c=hi,e^d=ehi,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, 6484027);