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 | |
4200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,11,5)(4,8,13,16)(10,12,15,14)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(33,34)(35,37)(36,41)(38,43)(39,42)(40,47)(44,48)(45,46);(1,3,6,11)(2,5,9,7)(4,10,13,15)(8,14,16,12)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35)(34,39)(36,44)(37,43)(38,42)(40,48)(41,46)(45,47);(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,36)(34,40)(35,44)(37,46)(38,47)(39,48)(41,43)(42,45);(1,3)(2,7)(4,10)(5,9)(6,11)(8,12)(13,15)(14,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,36)(34,41)(35,45)(37,46)(38,47)(39,48)(40,43)(42,44);(1,4)(2,8)(3,10)(5,14)(6,13)(7,12)(9,16)(11,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,35)(34,37)(36,44)(38,42)(39,43)(40,46)(41,48)(45,47);(1,5,6,7)(2,3,9,11)(4,12,13,14)(8,15,16,10)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,37)(34,42)(35,43)(36,46)(38,39)(40,45)(41,44)(47,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10365103);