#6607637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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

1515 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)(9,10)(11,16)(12,14)(13,15)(17,18,23,27)(19,24,29,39)(20,32,34,48)(21,26,22,37)(25,35,42,36)(28,43,33,47)(30,40,31,46)(38,44,41,45);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,29)(25,41)(26,40)(27,39)(28,34)(32,47)(35,45)(36,44)(37,46)(38,42)(43,48);(11,12)(14,16)(17,20,22,36,23,34,21,35)(18,25,26,32,27,42,37,48)(19,28,31,45,29,33,30,44)(24,38,40,43,39,41,46,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(19,29)(20,34)(24,39)(25,42)(30,31)(32,48)(35,36)(40,46);(11,12)(14,16)(17,21,23,22)(18,26,27,37)(19,30,29,31)(20,35,34,36)(24,40,39,46)(25,32,42,48)(28,44,33,45)(38,43,41,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=f,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=dh,e^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6514833);