#6485146

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469995
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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1401 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)(11,15)(12,16)(13,14)(17,18,23,27)(19,28,31,37)(20,25,22,35)(21,26,33,41)(24,36,39,29)(30,42,44,47)(32,40,34,45)(38,46,48,43);(9,10)(11,16)(12,15)(13,14)(17,19,23,31)(18,24,27,39)(20,29,22,36)(21,30,33,44)(25,37,35,28)(26,38,41,48)(32,43,34,46)(40,47,45,42);(2,5)(6,8)(10,14)(15,16)(17,20,23,22)(18,25,27,35)(19,29,31,36)(21,32,33,34)(24,37,39,28)(26,40,41,45)(30,43,44,46)(38,47,48,42);(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,30)(20,32)(22,34)(23,33)(24,38)(25,40)(27,41)(28,42)(29,43)(31,44)(35,45)(36,46)(37,47)(39,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(21,33)(26,41)(30,44)(32,34)(38,48)(40,45)(42,47)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6392342);