#6620451

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,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, 6527647);