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 | |
4165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18,23,27)(19,24,29,26)(20,30,28,25)(21,22,31,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,47)(43,44)(46,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19)(18,24)(20,31)(21,28)(22,25)(23,29)(26,27)(30,32)(33,35,38,39)(34,36,41,42)(37,43,47,46)(40,44,45,48);(2,6)(7,14)(8,11)(15,16)(17,20,23,28)(18,25,27,30)(19,21,29,31)(22,24,32,26)(33,35,38,39)(34,36,41,42)(37,46,47,43)(40,48,45,44);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,19)(18,26)(20,21)(22,25)(23,29)(24,27)(28,31)(30,32)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,21)(18,22)(19,28)(20,29)(23,31)(24,25)(26,30)(27,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,46)(43,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,22)(18,21)(19,25)(20,26)(23,32)(24,28)(27,31)(29,30)(33,37,38,47)(34,40,41,45)(35,43,39,46)(36,44,42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832234);