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 | |
4195 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)(3,8)(4,10)(5,7)(6,9)(11,15)(12,16)(13,14)(18,22)(19,25)(20,27)(31,32)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(2,5)(8,10)(9,13)(15,16)(17,18,21,22)(19,23,25,29)(20,24,27,30)(26,31,28,32)(33,35)(34,38)(36,42)(37,43)(39,41)(40,46)(44,47)(45,48);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,31)(27,28)(30,32)(33,34)(35,39)(36,40)(37,41)(38,43)(42,47)(44,45)(46,48);(1,4,7,3)(2,8,5,10)(6,11,14,12)(9,16,13,15)(17,19)(18,23)(20,28)(21,25)(22,29)(24,32)(26,27)(30,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,30)(28,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,20,21,27)(18,24,22,30)(19,26,25,28)(23,31,29,32)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gh,d^2=gh,e^2=gh,f^2=h,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, 10330001);