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 | |
3957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,12,9)(5,10,14,8)(11,16,13,15)(17,18,22,24)(19,26,21,23)(20,25,28,31)(27,32,29,30)(33,34,40,39)(35,44,42,37)(36,41,46,45)(38,43,47,48);(1,2,6,3)(4,7,12,9)(5,8,14,10)(11,15,13,16)(17,19,22,21)(18,23,24,26)(20,27,28,29)(25,30,31,32)(33,35)(34,37)(36,43)(38,41)(39,44)(40,42)(45,47)(46,48);(2,3)(7,9)(8,10)(15,16)(18,24)(23,26)(25,31)(30,32)(33,36,40,46)(34,41,39,45)(35,43,42,48)(37,38,44,47);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32)(33,37,40,44)(34,42,39,35)(36,38,46,47)(41,48,45,43);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,38)(34,43)(35,45)(36,44)(37,46)(39,48)(40,47)(41,42);(1,5)(2,8)(3,10)(4,11)(6,14)(7,15)(9,16)(12,13)(17,21,22,19)(18,26,24,23)(20,27,28,29)(25,30,31,32)(33,39,40,34)(35,44,42,37)(36,45,46,41)(38,43,47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=ghi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9758293);