#6589019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 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:

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

Polycyclic presentation:

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

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, 6496215);