#6451254

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1638 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)(9,10,13,15)(11,14,12,16)(17,18)(19,29)(20,26)(21,27)(22,25)(23,28)(24,33)(30,38)(31,45)(32,43)(34,48)(35,42)(36,40)(37,44)(39,47)(41,46);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,34)(21,32)(22,33)(23,30)(25,29)(26,41)(27,38)(28,43)(31,35)(36,46)(37,47)(39,42)(40,48)(44,45);(1,3,4,7)(2,6,5,8)(10,15)(14,16)(18,25)(19,30)(21,23)(24,38)(26,40)(29,43)(31,34)(32,33)(35,37)(39,48)(41,45)(46,47);(9,12,13,11)(10,16,15,14)(17,20,22,36)(18,26,25,40)(19,31,33,47)(21,35,23,37)(24,39,29,45)(27,42,28,44)(30,34,32,46)(38,48,43,41);(17,21,22,23)(18,27,25,28)(19,32,33,30)(20,35,36,37)(24,38,29,43)(26,42,40,44)(31,46,47,34)(39,48,45,41)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358450);