#6479639

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
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

1660 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)(19,28,31,39)(20,32,33,25)(21,26,22,36)(24,37,40,30)(29,43,44,42)(34,47,35,41)(38,45,46,48);(11,12)(14,16)(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,38)(26,39)(27,40)(28,36)(32,46)(33,44)(34,45)(35,48)(41,43)(42,47);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,23,33)(18,25,27,32)(19,29,31,44)(21,34,22,35)(24,38,40,46)(26,41,36,47)(28,42,39,43)(30,45,37,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(20,33)(25,32)(29,44)(34,35)(38,46)(41,47)(42,43)(45,48);(10,15)(14,16)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,34,33,35)(24,39,40,28)(25,41,32,47)(29,45,44,48)(38,43,46,42)

Polycyclic presentation:

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

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, 6386835);