#8720819

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8628015);