#6741697

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1432 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)(9,10)(11,15)(12,16)(13,14)(17,18,22,23)(19,27,30,24)(20,25,33,37)(21,34,36,26)(28,31,44,38)(29,39,46,42)(32,41,35,40)(43,48,45,47);(1,3)(2,5)(4,8)(6,7)(17,19,22,30)(18,24,23,27)(20,31,33,38)(21,29,36,46)(25,28,37,44)(26,39,34,42)(32,47,35,48)(40,43,41,45);(2,6)(5,7)(10,14)(15,16)(17,20,22,33)(18,25,23,37)(19,28,30,44)(21,35,36,32)(24,38,27,31)(26,41,34,40)(29,45,46,43)(39,47,42,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,36)(18,26,23,34)(19,29,30,46)(20,32,33,35)(24,39,27,42)(25,40,37,41)(28,43,44,45)(31,47,38,48);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(18,23)(19,24)(20,33)(26,34)(27,30)(28,31)(29,39)(32,35)(38,44)(42,46)(43,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fg,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6648893);