#6867767

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(3,7)(4,6)(5,8)(17,18,22,27,23,28,19,24)(20,31,29,40,33,42,37,25)(21,34,35,38,36,26,30,41)(32,45,43,39,47,44,46,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,22)(18,24,28,27)(20,29,33,37)(21,35,36,30)(25,31,40,42)(26,41,34,38)(32,46,47,43)(39,45,48,44);(10,14)(15,16)(17,20,23,33)(18,25,28,40)(19,29,22,37)(21,32,36,47)(24,31,27,42)(26,39,34,48)(30,43,35,46)(38,44,41,45);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,36)(18,26,28,34)(19,30,22,35)(20,32,33,47)(24,38,27,41)(25,39,40,48)(29,43,37,46)(31,44,42,45);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(21,36)(26,34)(30,35)(32,47)(38,41)(39,48)(43,46)(44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6774963);