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 | |
4158 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,31)(28,32)(33,34,39,37)(35,40,43,36)(38,41,47,45)(42,46,48,44);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,29)(30,32)(33,35,39,43)(34,36,37,40)(38,42,47,48)(41,44,45,46);(3,11)(4,12)(6,16)(8,13)(17,18,22,25)(19,23,27,20)(21,29,30,24)(26,28,32,31)(33,36,39,40)(34,35,37,43)(38,46,47,44)(41,48,45,42);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,18)(19,23)(20,27)(21,24)(22,25)(26,31)(28,32)(29,30)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,46,48,44);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32)(33,35)(34,40)(36,37)(38,42)(39,43)(41,46)(44,45)(47,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)(33,38,39,47)(34,41,37,45)(35,42,43,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835200);