#6859078

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(17,18,23,27)(19,24,22,36)(20,30,33,37)(21,26,35,42)(25,31,40,28)(29,38,34,41)(32,44,46,47)(39,45,48,43);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,23,22)(18,24,27,36)(20,31,33,28)(21,34,35,29)(25,30,40,37)(26,41,42,38)(32,43,46,45)(39,47,48,44);(10,14)(15,16)(17,20,23,33)(18,25,27,40)(19,28,22,31)(21,32,35,46)(24,37,36,30)(26,39,42,48)(29,43,34,45)(38,47,41,44);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,35)(18,26,27,42)(19,29,22,34)(20,32,33,46)(24,38,36,41)(25,39,40,48)(28,43,31,45)(30,44,37,47);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(21,35)(26,42)(29,34)(32,46)(38,41)(39,48)(43,45)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,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, 6766274);