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 | |
4153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,10,13,8)(4,5,14,9)(11,15,16,12)(17,18,23,27)(19,24,30,32)(20,22,31,25)(21,29,28,26)(33,34)(35,40)(36,39)(37,43)(38,41)(42,47)(44,46)(45,48);(2,7)(5,9)(8,10)(12,15)(17,19,23,30)(18,24,27,32)(20,28,31,21)(22,26,25,29)(33,34,39,36)(35,40,37,43)(38,44,46,41)(42,48,45,47);(1,3,6,13)(2,8,7,10)(4,11,14,16)(5,12,9,15)(17,20,23,31)(18,25,27,22)(19,21,30,28)(24,26,32,29)(33,35,39,37)(34,40,36,43)(38,42,46,45)(41,47,44,48);(1,3)(2,8)(4,11)(5,12)(6,13)(7,10)(9,15)(14,16)(17,20)(18,22)(19,28)(21,30)(23,31)(24,26)(25,27)(29,32)(33,36,39,34)(35,40,37,43)(38,44,46,41)(42,47,45,48);(1,4)(2,9)(3,11)(5,7)(6,14)(8,15)(10,12)(13,16)(17,21,23,28)(18,26,27,29)(19,20,30,31)(22,24,25,32)(33,37,39,35)(34,40,36,43)(38,45,46,42)(41,47,44,48);(1,5,6,9)(2,4,7,14)(3,12,13,15)(8,11,10,16)(17,22)(18,20)(19,29)(21,24)(23,25)(26,30)(27,31)(28,32)(33,38,39,46)(34,41,36,44)(35,42,37,45)(40,47,43,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662515);