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 | |
4141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,23,27)(19,28,29,24)(20,30,32,25)(21,22,26,31)(33,34)(35,40)(36,42)(37,38)(39,44)(41,46)(43,48)(45,47);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,15)(14,16)(17,19,23,29)(18,24,27,28)(20,31,32,22)(21,30,26,25)(33,35,39,45)(34,40,44,47)(36,37,46,43)(38,41,48,42);(2,6)(3,10)(4,13)(15,16)(20,32)(21,26)(22,31)(25,30)(33,36,39,46)(34,41,44,42)(35,43,45,37)(38,47,48,40);(1,3)(2,7)(4,14)(5,10)(6,9)(8,16)(11,13)(12,15)(17,20)(18,25)(19,22)(21,28)(23,32)(24,26)(27,30)(29,31)(33,37,39,43)(34,38,44,48)(35,36,45,46)(40,42,47,41);(1,4)(2,8)(3,11)(5,13)(6,12)(7,15)(9,16)(10,14)(17,21,23,26)(18,22,27,31)(19,25,29,30)(20,24,32,28)(33,36)(34,42)(35,37)(38,40)(39,46)(41,44)(43,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,22)(18,26)(19,20)(21,27)(23,31)(24,25)(28,30)(29,32)(33,38)(34,43)(35,41)(36,40)(37,44)(39,48)(42,45)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^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, 8965812);