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 | |
4106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,11,6)(4,12,13,8)(10,14,16,15)(17,18,22,25)(19,26,23,20)(21,24,30,32)(27,31,29,28)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,47,45,44);(1,2,5,9)(3,7,11,6)(4,12,13,8)(10,14,16,15)(17,19,22,23)(18,20,25,26)(21,29,30,27)(24,31,32,28)(33,35)(34,39)(36,41)(37,45)(38,43)(40,44)(42,46)(47,48);(1,3,5,11)(2,6,9,7)(4,10,13,16)(8,14,12,15)(17,18,22,25)(19,20,23,26)(21,24,30,32)(27,28,29,31)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,20)(18,23)(19,25)(21,28)(22,26)(24,29)(27,32)(30,31)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,46)(47,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,12)(11,16)(17,21,22,30)(18,24,25,32)(19,27,23,29)(20,28,26,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8757748);