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 | |
4196 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)(3,6)(4,11)(5,7)(8,13)(9,15)(10,16)(12,14)(17,18,23,25)(19,29,31,24)(20,32,30,26)(21,27,28,22)(33,34)(35,40)(36,37)(38,39)(41,44)(42,48)(43,45)(46,47);(1,3,5,9)(2,6,7,15)(4,12,13,10)(8,16,11,14)(17,19,23,31)(18,24,25,29)(20,28,30,21)(22,32,27,26)(34,39)(37,47)(40,41)(43,48);(2,7)(3,9)(8,11)(10,12)(18,25)(20,30)(21,28)(24,29)(33,35,38,44)(34,40,39,41)(36,45,46,42)(37,43,47,48);(1,2)(3,6)(4,8)(5,7)(9,15)(10,14)(11,13)(12,16)(17,20,23,30)(18,26,25,32)(19,28,31,21)(22,29,27,24)(33,35)(34,41)(36,42)(37,43)(38,44)(39,40)(45,46)(47,48);(1,3)(2,6)(4,10)(5,9)(7,15)(8,14)(11,16)(12,13)(17,21,23,28)(18,27,25,22)(19,30,31,20)(24,32,29,26)(33,36)(34,37)(35,42)(38,46)(39,47)(40,48)(41,43)(44,45);(1,4,5,13)(2,8,7,11)(3,10,9,12)(6,14,15,16)(17,22,23,27)(18,28,25,21)(19,26,31,32)(20,29,30,24)(33,37)(34,36)(35,43)(38,47)(39,46)(40,45)(41,42)(44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8714097);