#6483038

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1428 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)(19,31)(20,36)(21,27)(22,28)(23,29)(24,30)(25,34)(26,37)(32,45)(33,44)(35,43)(38,42)(39,46)(40,47)(41,48);(2,5)(6,8)(10,14)(11,12)(17,19,23,34,24,35,22,33)(18,25,28,31,30,44,29,43)(20,21,32,39,38,41,37,40)(26,27,42,47,45,48,36,46);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20,23,32,24,38,22,37)(18,26,28,42,30,45,29,36)(19,21,34,39,35,41,33,40)(25,27,31,47,44,48,43,46);(9,12,13,11)(10,16,14,15)(17,21)(18,27)(19,32)(20,33)(22,40)(23,39)(24,41)(25,42)(26,43)(28,47)(29,46)(30,48)(31,45)(34,38)(35,37)(36,44);(10,14)(15,16)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,32,38,37)(21,39,41,40)(25,31,44,43)(26,36,45,42)(27,46,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6390234);