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 | |
4123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,9,10)(4,12,13,8)(11,16,15,14)(17,18,22,23)(19,21,26,24)(20,25,28,31)(27,30,29,32)(33,34)(35,40)(36,42)(37,43)(38,46)(39,41)(44,48)(45,47);(2,6)(7,10)(8,12)(14,16)(18,23)(19,26)(20,28)(30,32)(33,35)(34,40)(36,39)(37,44)(38,45)(41,42)(43,48)(46,47);(2,6)(3,9)(4,13)(14,16)(19,26)(20,28)(21,24)(25,31)(34,41)(35,36)(37,47)(38,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,26)(18,24,23,21)(20,29,28,27)(25,30,31,32)(33,36,39,35)(34,42,41,40)(37,44,47,46)(38,43,48,45);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20,22,28)(18,25,23,31)(19,27,26,29)(21,30,24,32)(33,37,39,47)(34,43,41,45)(35,44,36,46)(38,40,48,42);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,21,22,24)(18,26,23,19)(20,30,28,32)(25,29,31,27)(33,38)(34,44)(35,45)(36,43)(37,40)(39,48)(41,46)(42,47)Polycyclic presentation:
2^9|a^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966305);