#6620473

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,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, 6527669);