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 | |
3942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,11)(4,8)(5,6)(7,10)(9,16)(12,15)(13,14)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,47,45);(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35,38,43)(34,39,41,36)(37,42,46,47)(40,45,48,44);(1,3,7,4)(2,8,10,11)(5,12,9,14)(6,13,16,15)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,45,47,44);(1,4,7,3)(2,8,10,11)(5,12,9,14)(6,15,16,13)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,36)(34,35)(37,45)(38,39)(40,47)(41,43)(42,48)(44,46);(1,5,7,9)(2,6,10,16)(3,12,4,14)(8,13,11,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,45)(44,48)(46,47);(1,6)(2,9)(3,13)(4,15)(5,10)(7,16)(8,14)(11,12)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,45)(41,48)(43,47)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,d^2=hi,e^2=hi,f^2=i,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, 10358790);