#6471370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,21,26)(20,31,34,44)(22,36,38,27)(25,37,42,33)(29,45,32,48)(30,43,35,40)(39,47,41,46);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,23)(22,30)(25,39)(26,28)(27,40)(31,45)(32,34)(33,46)(35,38)(36,43)(37,47)(41,42)(44,48);(11,15)(13,16)(17,20,23,34)(18,25,28,42)(19,29,21,32)(22,37,38,33)(24,39,26,41)(27,44,36,31)(30,47,35,46)(40,48,43,45);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,21,23,19)(18,26,28,24)(20,32,34,29)(22,35,38,30)(25,41,42,39)(27,43,36,40)(31,48,44,45)(33,47,37,46);(10,14)(13,16)(19,21)(24,26)(29,32)(30,35)(39,41)(40,43)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,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, 6378566);