#6452683

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1663 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)(11,16)(12,14)(13,15)(17,18,22,27)(19,29,33,24)(20,34,36,25)(21,26,23,28)(30,39,47,45)(31,41,32,40)(35,42,46,48)(37,44,38,43);(2,5)(6,8)(17,19,22,33)(18,24,27,29)(20,35,36,46)(21,32,23,31)(25,42,34,48)(26,41,28,40)(30,38,47,37)(39,44,45,43);(9,11,13,12)(10,14,15,16)(17,20,22,36)(18,25,27,34)(19,30,33,47)(21,37,23,38)(24,39,29,45)(26,43,28,44)(31,35,32,46)(40,42,41,48);(9,12,13,11)(10,14,15,16)(19,31)(20,36)(21,23)(24,40)(25,34)(26,28)(29,41)(30,46)(32,33)(35,47)(39,48)(42,45);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,27,28)(19,32,33,31)(20,37,36,38)(24,41,29,40)(25,43,34,44)(30,46,47,35)(39,48,45,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,h^c=hi,e^d=ei,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, 6359879);