#6744173

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

1280 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)(19,23)(20,25)(21,24)(22,26)(27,40)(28,35)(29,36)(30,34)(31,37)(32,39)(33,38)(41,46)(42,45)(43,47)(44,48);(10,13)(11,16)(17,19,21,29)(18,23,24,36)(20,28,31,43)(22,27,33,30)(25,35,37,47)(26,40,38,34)(32,41,44,42)(39,46,48,45);(2,6)(7,8)(9,11,12,16)(10,14,13,15)(18,24)(19,27)(20,31)(22,33)(23,34)(28,42)(29,30)(35,46)(36,40)(39,48)(41,43)(45,47);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,21,31)(18,25,24,37)(19,28,29,43)(22,32,33,44)(23,35,36,47)(26,39,38,48)(27,41,30,42)(34,45,40,46);(1,4,5,3)(2,8,6,7)(18,24)(19,27)(22,33)(23,34)(25,37)(28,41)(29,30)(32,44)(35,45)(36,40)(42,43)(46,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,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,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, 6651369);