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 | |
4227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,11,13,7)(4,14,15,9)(5,10,16,12)(17,18,23,21)(19,26,20,24)(22,29,30,25)(27,31,28,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(1,3,6,13)(2,7,8,11)(4,10,15,12)(5,9,16,14)(19,20)(22,30)(24,26)(25,29)(35,43)(37,47)(39,48)(41,44);(2,8)(5,16)(7,11)(9,14)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,4,6,15)(2,9,8,14)(3,12,13,10)(5,11,16,7)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,5)(2,10)(3,9)(4,11)(6,16)(7,15)(8,12)(13,14)(17,21,23,18)(19,24,20,26)(22,29,30,25)(27,31,28,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,3)(2,7)(4,12)(5,9)(6,13)(8,11)(10,15)(14,16)(17,22,23,30)(18,25,21,29)(19,27,20,28)(24,31,26,32)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 10235384);