#6742521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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

1512 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,21)(19,24,30,28)(20,32,34,25)(23,26,37,36)(27,45,46,39)(29,40,31,41)(33,48,47,42)(35,44,38,43);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,33)(21,28)(22,30)(23,29)(25,42)(26,40)(27,35)(31,37)(32,48)(34,47)(36,41)(38,46)(39,43)(44,45);(10,14)(13,15)(17,20,22,34)(18,25,21,32)(19,27,30,46)(23,38,37,35)(24,39,28,45)(26,44,36,43)(29,33,31,47)(40,42,41,48);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,21,22,18)(19,28,30,24)(20,25,34,32)(23,36,37,26)(27,39,46,45)(29,41,31,40)(33,42,47,48)(35,43,38,44);(1,5,6,4)(2,7,3,8)(11,16)(13,15)(19,29)(23,37)(24,40)(26,36)(27,33)(28,41)(30,31)(35,38)(39,42)(43,44)(45,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6649717);