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 | |
4030 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,24)(19,22,20,27)(21,25,30,32)(26,31,28,29)(33,34,38,41)(35,39,43,36)(37,45,47,40)(42,44,48,46);(4,11)(8,14)(10,15)(12,16)(17,19)(18,22)(20,23)(21,26)(24,27)(25,31)(28,30)(29,32)(33,35,38,43)(34,36,41,39)(37,42,47,48)(40,44,45,46);(2,7)(3,9)(8,14)(10,15)(18,24)(19,20)(25,32)(26,28)(33,35,38,43)(34,36,41,39)(37,42,47,48)(40,44,45,46);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,19)(18,22,24,27)(21,28,30,26)(25,31,32,29)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,25)(19,26)(20,28)(22,31)(23,30)(24,32)(27,29)(33,36,38,39)(34,35,41,43)(37,46,47,44)(40,48,45,42);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,19)(20,24)(21,29)(23,27)(25,28)(26,32)(30,31)(33,37,38,47)(34,40,41,45)(35,42,43,48)(36,44,39,46)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9843594);