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 | |
4152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(4,12)(7,10)(8,13)(17,18)(19,26)(20,24)(21,29)(22,23)(25,31)(27,28)(30,32)(33,34,39,43)(35,40,44,47)(36,45,41,37)(38,46,48,42);(1,2)(3,10)(4,13)(5,6)(7,9)(8,12)(11,15)(14,16)(17,19,23,24)(18,20,22,26)(21,27,31,32)(25,28,29,30)(33,35)(34,40)(36,46)(37,38)(39,44)(41,42)(43,47)(45,48);(2,6)(3,9)(8,13)(11,14)(18,22)(20,26)(25,29)(28,30)(33,36,39,41)(34,37,43,45)(35,42,44,46)(38,40,48,47);(1,3)(2,7)(4,14)(5,9)(6,10)(8,16)(11,12)(13,15)(17,20,23,26)(18,24,22,19)(21,30,31,28)(25,27,29,32)(33,36)(34,37)(35,42)(38,47)(39,41)(40,48)(43,45)(44,46);(1,4)(2,8)(3,11)(5,12)(6,13)(7,15)(9,14)(10,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,31)(24,32)(26,30)(33,37)(34,41)(35,38)(36,43)(39,45)(40,42)(44,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,22,23,18)(19,26,24,20)(21,29,31,25)(27,30,32,28)(33,38)(34,42)(35,37)(36,47)(39,48)(40,41)(43,46)(44,45)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8840280);