#6516147

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 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)(17,18,22,27)(19,28,23,24)(20,31,33,44)(21,26,36,43)(25,37,40,29)(30,41,34,38)(32,47,35,48)(39,46,42,45);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,27,28)(20,29,33,37)(21,34,36,30)(25,31,40,44)(26,41,43,38)(32,46,35,45)(39,48,42,47);(10,13)(11,15)(17,20,22,33)(18,25,27,40)(19,29,23,37)(21,35,36,32)(24,31,28,44)(26,42,43,39)(30,46,34,45)(38,48,41,47);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,30)(20,32)(22,36)(23,34)(24,38)(25,39)(27,43)(28,41)(29,45)(31,47)(33,35)(37,46)(40,42)(44,48);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(21,36)(26,43)(30,34)(32,35)(38,41)(39,42)(45,46)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6423343);