#6623995

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(17,18,22,27,23,28,21,26)(19,29,32,40,33,39,31,24)(20,34,35,41,36,42,37,25)(30,38,44,47,45,48,46,43);(2,5)(6,8)(10,14)(12,15)(17,19,23,33)(18,24,28,40)(20,30,36,45)(21,32,22,31)(25,38,41,48)(26,29,27,39)(34,43,42,47)(35,46,37,44);(9,10)(11,12)(13,14)(15,16)(17,20,23,36)(18,25,28,41)(19,30,33,45)(21,35,22,37)(24,38,40,48)(26,34,27,42)(29,43,39,47)(31,44,32,46);(1,3,4,7)(2,6,5,8)(9,11,13,16)(10,15,14,12);(9,12,13,15)(10,11,14,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,34,41,42)(30,44,45,46)(38,47,48,43)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531191);