#6434484

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1676 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)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,32,40)(20,34,37,41)(21,26,22,27)(24,31,39,33)(25,36,42,35)(30,44,46,38)(43,45,48,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,24,28,39)(20,30,37,46)(21,31,22,33)(25,38,42,44)(26,29,27,40)(34,48,41,43)(35,45,36,47);(11,15)(13,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,46)(33,47)(39,44)(40,48);(10,14)(13,16)(17,21,23,22)(18,26,28,27)(19,31,32,33)(20,35,37,36)(24,29,39,40)(25,41,42,34)(30,45,46,47)(38,43,44,48);(1,3,4,7)(2,6,5,8)(19,32)(24,39)(29,40)(30,46)(31,33)(38,44)(43,48)(45,47)

Polycyclic presentation:

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

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, 6341680);