#6744190

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 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)(4,8)(5,6)(9,10,12,13)(11,15,16,14)(17,18,22,25)(19,23,30,37)(20,24,31,38)(21,26,33,40)(27,34,29,36)(28,35,43,47)(32,39,44,48)(41,45,42,46);(10,13)(11,16)(17,19,22,30)(18,23,25,37)(20,28,31,43)(21,27,33,29)(24,35,38,47)(26,34,40,36)(32,41,44,42)(39,45,48,46);(2,6)(7,8)(9,11,12,16)(10,14,13,15)(19,27)(20,31)(21,33)(23,34)(24,38)(26,40)(28,42)(29,30)(35,46)(36,37)(41,43)(45,47);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,22,31)(18,24,25,38)(19,28,30,43)(21,32,33,44)(23,35,37,47)(26,39,40,48)(27,41,29,42)(34,45,36,46);(1,4,5,3)(2,8,6,7)(18,25)(19,27)(21,33)(23,36)(24,38)(28,41)(29,30)(32,44)(34,37)(35,46)(42,43)(45,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,d^2=fgi,e^2=fg,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,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6651386);