#6488930

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470402
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1322 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)(11,14)(12,15)(13,16)(17,18,23,28)(19,29,34,46)(20,25,35,43)(21,26,36,45)(22,38,30,27)(24,39,42,33)(31,44,47,41)(32,40,37,48);(11,12)(14,15)(17,19,30,39,23,34,22,33)(18,24,38,46,28,42,27,29)(20,31,21,37,35,47,36,32)(25,40,45,41,43,48,26,44);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20,23,35)(18,25,28,43)(19,31,34,47)(21,22,36,30)(24,40,42,48)(26,38,45,27)(29,44,46,41)(32,39,37,33);(9,12,13,11)(10,15,16,14)(17,21,23,36)(18,26,28,45)(19,32,34,37)(20,30,35,22)(24,41,42,44)(25,27,43,38)(29,40,46,48)(31,33,47,39);(1,3,4,7)(2,6,5,8)(10,16)(14,15)(20,35)(21,36)(25,43)(26,45)(31,47)(32,37)(40,48)(41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6396126);