Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,4)(3,9)(8,12)(11,13)(17,18,23,27)(19,28,29,24)(20,25,30,32)(21,22,26,31)(34,40)(35,44)(36,46)(43,47);(1,2,6,4)(3,7,9,10)(5,8,14,12)(11,15,13,16)(19,29)(20,30)(24,28)(25,32)(33,34)(35,41)(36,38)(37,43)(39,40)(42,46)(44,48)(45,47);(17,19)(18,24)(20,26)(21,30)(22,25)(23,29)(27,28)(31,32)(33,35)(34,41)(36,45)(37,46)(38,47)(39,44)(40,48)(42,43);(1,3)(2,7)(4,10)(5,13)(6,9)(8,16)(11,14)(12,15)(17,20,23,30)(18,25,27,32)(19,26,29,21)(22,28,31,24)(33,36,39,46)(34,42,40,38)(35,45,44,37)(41,43,48,47);(1,4,6,2)(3,10,9,7)(5,12,14,8)(11,16,13,15)(17,21,23,26)(18,22,27,31)(19,30,29,20)(24,25,28,32)(33,37,39,45)(34,43,40,47)(35,46,44,36)(38,41,42,48);(1,5,6,14)(2,8,4,12)(3,11,9,13)(7,15,10,16)(17,22)(18,26)(19,25)(20,24)(21,27)(23,31)(28,30)(29,32)(33,38,39,42)(34,36,40,46)(35,47,44,43)(37,41,45,48)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8353079);