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 | |
4235 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,11,7)(4,12,14,8)(10,15,13,16)(17,18)(19,24)(20,30)(21,31)(22,25)(23,29)(26,32)(27,28)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,47,45);(2,6)(7,9)(8,12)(15,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,25)(26,31)(30,32)(33,35,38,43)(34,39,41,36)(37,42,46,47)(40,45,48,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,20)(18,21)(19,26)(22,31)(23,32)(24,28)(25,30)(27,29)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,45,47,44);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,14)(12,15)(17,20)(18,21)(19,27)(22,31)(23,28)(24,32)(25,30)(26,29)(33,36)(34,35)(37,45)(38,39)(40,47)(41,43)(42,48)(44,46);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,19)(18,24)(20,26)(21,28)(22,29)(23,25)(27,31)(30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,45)(44,48)(46,47);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,21)(18,20)(19,28)(22,30)(23,27)(24,26)(25,31)(29,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,45)(41,48)(43,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10319422);