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 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,10,8)(4,5,13,14)(11,15,16,12)(17,18)(19,20)(21,29)(22,23)(24,31)(25,28)(26,32)(27,30)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(2,7)(3,10)(5,14)(11,16)(17,19,22,25)(18,20,23,28)(21,30,31,26)(24,32,29,27)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(2,7)(4,13)(8,9)(11,16)(18,23)(19,25)(24,29)(26,30)(37,45)(40,44)(42,48)(46,47);(1,3,6,10)(2,8,7,9)(4,11,13,16)(5,12,14,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,30)(31,32)(33,36,38,34)(35,39,43,41)(37,40,45,44)(42,47,48,46);(1,4)(2,5)(3,11)(6,13)(7,14)(8,12)(9,15)(10,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48);(1,5)(2,4)(3,12)(6,14)(7,13)(8,11)(9,16)(10,15)(17,21)(18,24)(19,26)(20,27)(22,31)(23,29)(25,30)(28,32)(33,35)(34,41)(36,39)(37,42)(38,43)(40,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=hi,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, 10231803);