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 | |
4007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,18,24,27)(19,25,22,23)(20,30,21,26)(28,31,29,32)(33,34,39,44)(35,45,47,40)(36,37,42,41)(38,43,46,48);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,10)(15,16)(17,19,24,22)(18,25,27,23)(20,29,21,28)(26,31,30,32)(33,35)(34,40)(36,46)(37,43)(38,42)(39,47)(41,48)(44,45);(17,20,24,21)(18,26,27,30)(19,28,22,29)(23,31,25,32)(33,36,39,42)(34,41,44,37)(35,46,47,38)(40,48,45,43);(1,4,7,13)(2,9,6,15)(3,12,14,5)(8,10,11,16)(17,21,24,20)(18,26,27,30)(19,29,22,28)(23,31,25,32)(33,34)(35,45)(36,41)(37,42)(38,48)(39,44)(40,47)(43,46);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,9,11,15)(17,22,24,19)(18,23,27,25)(20,29,21,28)(26,31,30,32)(33,37,39,41)(34,42,44,36)(35,43,47,48)(38,45,46,40);(1,6,7,2)(3,11,14,8)(4,9,13,15)(5,16,12,10)(17,23)(18,19)(20,31)(21,32)(22,27)(24,25)(26,28)(29,30)(33,38,39,46)(34,43,44,48)(35,42,47,36)(37,45,41,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=h,d^2=gh,e^2=gi,f^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, 8353150);