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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,4,8)(5,13,14,10)(11,15,12,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34,39,43)(35,42,46,40)(36,38,37,41)(44,45,47,48);(2,7)(8,9)(10,13)(15,16)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35)(34,40)(36,47)(37,44)(38,45)(39,46)(41,48)(42,43);(1,3,6,4)(2,8,7,9)(5,11,14,12)(10,15,13,16)(17,19,22,20)(18,24,26,23)(21,27,29,28)(25,31,32,30)(33,36,39,37)(34,41,43,38)(35,44,46,47)(40,45,42,48);(1,4,6,3)(2,9,7,8)(5,11,14,12)(10,15,13,16)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)(33,37,39,36)(34,38,43,41)(35,44,46,47)(40,45,42,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,35)(34,42)(36,44)(37,47)(38,45)(39,46)(40,43)(41,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,38)(34,36)(35,45)(37,43)(39,41)(40,47)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10317254);