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 | |
4113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34,39,38)(35,40,42,36)(37,45,47,41)(43,44,48,46);(2,7)(3,12)(9,14)(10,13)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35)(34,40)(36,38)(37,43)(39,42)(41,46)(44,45)(47,48);(2,7)(5,16)(8,11)(9,14)(21,29)(24,31)(26,32)(28,30)(34,38)(35,42)(41,45)(43,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,36)(34,35)(37,46)(38,42)(39,40)(41,48)(43,45)(44,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,37,39,47)(34,41,38,45)(35,43,42,48)(36,44,40,46);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,38,39,34)(35,36,42,40)(37,45,47,41)(43,44,48,46)Polycyclic presentation:
2^9|a^2=gh,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229398);