#6620174

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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1702 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:

(1,2,5,6)(3,7,4,8)(9,10,13,14)(11,15,12,16)(17,18)(19,24)(20,32)(21,27)(22,35)(23,26)(25,33)(28,42)(29,39)(30,38)(31,45)(34,41)(36,40)(37,44)(43,47)(46,48);(2,6)(7,8)(10,14)(11,12)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,42)(33,44)(34,43)(35,45)(36,46)(40,48)(41,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,23,34)(18,25,26,40)(19,28,29,43)(21,33,22,36)(24,37,39,48)(27,41,35,32)(30,44,31,46)(38,47,45,42);(1,4,5,3)(2,8,6,7)(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=f,c^2=fi,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, 6527370);