#6589272

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,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, 6496468);