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 | |
4302 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)(3,11,12,8)(4,9,10,5)(13,14,16,15)(18,22)(23,26)(24,28)(30,32)(33,34)(35,36)(37,45)(38,39)(40,47)(41,44)(42,48)(43,46);(2,7)(3,12)(5,9)(13,16)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30)(33,35,38,41)(34,36,39,44)(37,46,47,42)(40,48,45,43);(2,7)(4,10)(8,11)(13,16)(19,25)(23,26)(27,31)(30,32)(34,39)(35,41)(40,45)(42,46);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,15)(10,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,46)(47,48);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,15,11,14)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,34)(35,36)(37,40)(38,39)(41,44)(42,43)(45,47)(46,48);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,16,11,13)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,37)(34,40)(35,42)(36,43)(38,47)(39,45)(41,46)(44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,e^2=g,f^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, 10231843);