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 | |
4156 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)(19,22)(20,21)(23,25)(24,27)(26,31)(28,29)(30,32)(33,34,38,41)(35,39,43,37)(36,44,47,40)(42,46,45,48);(1,2)(3,10)(4,13)(5,6)(7,9)(8,12)(11,15)(14,16)(17,19,23,27)(18,24,25,22)(20,30,26,28)(21,29,31,32)(33,35,38,43)(34,37,41,39)(36,45,47,42)(40,48,44,46);(2,6)(3,9)(8,13)(11,14)(18,25)(20,26)(22,24)(28,30)(33,35,38,43)(34,39,41,37)(36,42,47,45)(40,48,44,46);(1,3)(2,7)(4,14)(5,9)(6,10)(8,16)(11,12)(13,15)(17,20)(18,21)(19,28)(22,29)(23,26)(24,32)(25,31)(27,30)(33,35)(34,39)(36,45)(37,41)(38,43)(40,46)(42,47)(44,48);(1,4)(2,8)(3,11)(5,12)(6,13)(7,15)(9,14)(10,16)(17,21)(18,26)(19,29)(20,25)(22,30)(23,31)(24,28)(27,32)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,44)(43,45);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,22,23,24)(18,27,25,19)(20,29,26,32)(21,30,31,28)(33,37,38,39)(34,35,41,43)(36,46,47,48)(40,42,44,45)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,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, 8840272);