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 | |
4144 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,15)(12,16)(18,22)(23,26)(24,28)(30,32)(33,34,39,40)(35,38,43,46)(36,41,45,47)(37,42,44,48);(2,7)(3,10)(5,14)(12,15)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30)(33,35,39,43)(34,38,40,46)(36,44,45,37)(41,48,47,42);(2,7)(4,8)(9,13)(12,15)(19,25)(23,26)(27,31)(30,32)(34,40)(35,43)(37,44)(41,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,13)(14,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36)(34,41)(35,37)(38,42)(39,45)(40,47)(43,44)(46,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,37)(34,42)(35,36)(38,41)(39,44)(40,48)(43,45)(46,47);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,38)(34,35)(36,42)(37,41)(39,46)(40,43)(44,47)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231781);