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 | |
4260 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)(3,10)(4,8)(5,13)(6,7)(9,14)(11,12)(15,16)(17,18,23,24)(19,22,27,30)(20,25,29,31)(21,26,28,32)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(2,7)(3,8)(5,14)(12,16)(17,19,23,27)(18,22,24,30)(20,28,29,21)(25,32,31,26)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(2,7)(4,10)(9,13)(12,16)(18,24)(19,27)(21,28)(25,31)(37,45)(40,44)(42,48)(46,47);(1,3,6,8)(2,4,7,10)(5,11,14,15)(9,12,13,16)(17,20)(18,25)(19,21)(22,26)(23,29)(24,31)(27,28)(30,32)(33,36,38,34)(35,39,43,41)(37,40,45,44)(42,47,48,46);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,21)(18,26)(19,20)(22,25)(23,28)(24,32)(27,29)(30,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,22)(18,19)(20,26)(21,25)(23,30)(24,27)(28,31)(29,32)(33,35)(34,41)(36,39)(37,42)(38,43)(40,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231741);