#6821985

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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

1301 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)(11,14)(12,13)(15,16)(17,18)(19,25)(20,34)(21,27)(22,39)(23,29)(24,30)(26,37)(28,32)(31,47)(33,42)(35,45)(36,46)(38,44)(40,43)(41,48);(1,3)(2,5)(4,8)(6,7)(17,19,23,33)(18,25,29,42)(20,35,37,40)(21,32,24,22)(26,43,34,45)(27,28,30,39)(31,36,48,38)(41,44,47,46);(10,13)(14,16)(17,20,21,36,23,37,24,38)(18,26,27,44,29,34,30,46)(19,31,22,40,33,48,32,35)(25,41,39,45,42,47,28,43);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,27,29,30)(19,32,33,22)(20,36,37,38)(25,28,42,39)(26,44,34,46)(31,35,48,40)(41,43,47,45);(3,8)(5,7)(17,22,24,19,23,32,21,33)(18,28,30,42,29,39,27,25)(20,35,38,48,37,40,36,31)(26,45,46,41,34,43,44,47)

Polycyclic presentation:

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

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, 6729181);