#6555875

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1512 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=i,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6463071);