#6507324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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	#24271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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)(3,6)(4,5)(7,8)(17,18,22,27)(19,28,30,24)(20,32,33,47)(21,26,23,37)(25,34,41,35)(29,43,45,46)(31,40,36,39)(38,44,42,48);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,40)(27,28)(32,46)(33,45)(34,48)(35,44)(37,39)(41,42)(43,47);(9,11)(10,13)(12,16)(14,15)(17,20,23,35,22,33,21,34)(18,25,26,32,27,41,37,47)(19,29,31,44,30,45,36,48)(24,38,40,46,28,42,39,43);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(19,30)(20,33)(24,28)(25,41)(31,36)(32,47)(34,35)(39,40);(10,14)(13,15)(17,21,22,23)(18,26,27,37)(19,31,30,36)(20,34,33,35)(24,39,28,40)(25,32,41,47)(29,44,45,48)(38,43,42,46)

Polycyclic presentation:

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

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, 6414520);