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 | |
4002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,8,11,7)(5,9,13,12)(10,15,16,14)(19,25)(24,31)(26,29)(27,32)(33,34,38,39)(35,41,43,36)(37,45,46,40)(42,44,48,47);(1,3,6,11)(2,7,4,8)(5,10,13,16)(9,14,12,15)(17,18,22,23)(19,24,25,31)(20,28,30,21)(26,27,29,32)(34,39)(36,41)(40,45)(44,47);(5,13)(9,12)(10,16)(14,15)(18,23)(21,28)(24,31)(27,32)(33,34)(35,36)(37,40)(38,39)(41,43)(42,44)(45,46)(47,48);(1,4,6,2)(3,8,11,7)(5,12,13,9)(10,15,16,14)(17,19)(18,24)(20,29)(21,27)(22,25)(23,31)(26,30)(28,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,47)(46,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,20)(18,21)(19,26)(22,30)(23,28)(24,32)(25,29)(27,31)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,48)(46,47);(1,5,6,13)(2,9,4,12)(3,10,11,16)(7,14,8,15)(17,21)(18,20)(19,27)(22,28)(23,30)(24,29)(25,32)(26,31)(33,37,38,46)(34,40,39,45)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833496);