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 | |
4102 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)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,18,24,30)(19,27,21,25)(20,29,23,26)(22,28,32,31)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,14)(15,16)(17,19,24,21)(18,25,30,27)(20,31,23,28)(22,29,32,26)(36,44)(39,46)(42,48)(45,47);(1,2,5,8)(3,9,11,6)(4,7,13,12)(10,15,16,14)(17,20,24,23)(18,26,30,29)(19,31,21,28)(22,27,32,25)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,21,24,19)(18,27,30,25)(20,31,23,28)(22,29,32,26);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,22,24,32)(18,28,30,31)(19,26,21,29)(20,25,23,27)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,23,24,20)(18,29,30,26)(19,31,21,28)(22,27,32,25)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=h,d^2=g,e^2=g,f^2=g,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, 8669247);