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 | |
4174 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(17,18)(19,23)(20,21)(22,25)(24,28)(26,31)(27,29)(30,32)(36,44)(40,45)(43,48)(46,47);(2,7)(4,14)(8,11)(10,16)(17,19,22,28)(18,23,25,24)(20,29,30,26)(21,27,32,31)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,46)(47,48);(17,19)(18,24)(20,26)(21,27)(22,28)(23,25)(29,30)(31,32)(33,35)(34,37)(36,43)(38,41)(39,42)(40,46)(44,48)(45,47);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,13,15,9)(17,19)(18,23)(20,26)(21,31)(22,28)(24,25)(27,32)(29,30)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,48)(45,46);(1,4)(2,9)(3,10)(5,8)(6,14)(7,13)(11,15)(12,16)(17,20)(18,21)(19,26)(22,30)(23,31)(24,27)(25,32)(28,29)(33,36,38,44)(34,40,42,45)(35,43,41,48)(37,46,39,47);(1,5)(2,10)(3,9)(4,8)(6,15)(7,16)(11,14)(12,13)(17,21)(18,20)(19,27)(22,32)(23,29)(24,26)(25,30)(28,31)(33,37,38,39)(34,41,42,35)(36,46,44,47)(40,48,45,43)Polycyclic presentation:
2^9|b^2=h,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8413503);