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 | |
4556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,16)(4,14,12,6)(5,9,15,13)(17,18,22,23)(19,25,27,20)(21,24,30,31)(26,29,32,28)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,45)(46,47);(1,3)(2,8)(4,5)(6,13)(7,11)(9,14)(10,16)(12,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(34,39)(36,40)(42,47)(44,48);(3,11)(4,12)(6,14)(8,16)(18,23)(20,25)(24,31)(28,29)(33,35,38,41)(34,40,39,36)(37,45,46,43)(42,44,47,48);(1,4)(2,6)(3,5)(7,12)(8,13)(9,16)(10,14)(11,15)(17,20)(18,19)(21,29)(22,25)(23,27)(24,32)(26,31)(28,30)(33,35)(34,40)(36,39)(37,43)(38,41)(42,48)(44,47)(45,46);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,13)(14,16)(17,21,22,30)(18,24,23,31)(19,26,27,32)(20,28,25,29)(33,36,38,40)(34,41,39,35)(37,44,46,48)(42,45,47,43);(1,6)(2,4)(3,13)(5,8)(7,14)(9,11)(10,12)(15,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31)(33,37)(34,42)(35,43)(36,44)(38,46)(39,47)(40,48)(41,45)Polycyclic presentation:
2^9|a^2=hi,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835876);