#7063394

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1484 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,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, 6970590);