#6580225

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1517 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,d^2=f,e^2=i,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, 6487421);