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 | |
3976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,16)(4,8,14,13)(5,9,15,12)(17,18,22,23)(19,20,26,27)(21,24,29,30)(25,28,31,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,46)(42,47)(45,48);(3,11)(5,15)(7,16)(9,12)(18,23)(20,27)(21,29)(25,31)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,3)(2,7)(4,9)(5,13)(6,11)(8,15)(10,16)(12,14)(17,19,22,26)(18,20,23,27)(21,25,29,31)(24,28,30,32)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(1,3)(2,7)(4,9)(5,13)(6,11)(8,15)(10,16)(12,14)(17,20,22,27)(18,19,23,26)(21,28,29,32)(24,25,30,31)(33,35)(34,38)(36,42)(37,43)(39,47)(40,41)(44,48)(45,46);(1,4)(2,8)(3,12)(5,7)(6,14)(9,11)(10,13)(15,16)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,31)(30,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,46)(45,48);(1,5)(2,9)(3,13)(4,7)(6,15)(8,11)(10,12)(14,16)(17,21,22,29)(18,24,23,30)(19,25,26,31)(20,28,27,32)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=gh,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, 10232334);