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 | |
4277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,11)(10,15)(20,28)(24,31)(27,32)(29,30)(33,34,39,44)(35,45,47,40)(36,37,42,41)(38,43,48,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,35,39,47)(34,40,44,45)(36,38,42,48)(37,46,41,43);(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,39,42)(34,41,44,37)(35,38,47,48)(40,43,45,46);(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,34)(35,45)(36,41)(37,42)(38,46)(39,44)(40,47)(43,48);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,37,39,41)(34,42,44,36)(35,46,47,43)(38,40,48,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,38)(34,43)(35,42)(36,47)(37,40)(39,48)(41,45)(44,46)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8341305);