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 | |
4212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,6,10)(5,9,14,12)(11,15,13,16)(17,18,24,27)(19,28,30,25)(20,22,21,31)(23,26,32,29)(33,34)(35,43)(36,38)(37,41)(39,40)(42,46)(44,45)(47,48);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,19)(18,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32)(34,40)(36,46)(41,43)(44,47);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,20,24,21)(18,22,27,31)(19,26,30,29)(23,28,32,25)(33,35,39,37)(34,41,40,43)(36,44,46,47)(38,48,42,45);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,21,24,20)(18,22,27,31)(19,29,30,26)(23,28,32,25)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,45)(43,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22)(18,21)(19,23)(20,27)(24,31)(25,29)(26,28)(30,32)(33,37,39,35)(34,43,40,41)(36,44,46,47)(38,48,42,45);(1,6,7,3)(2,10,4,8)(5,13,14,11)(9,16,12,15)(17,23)(18,26)(19,22)(20,25)(21,28)(24,32)(27,29)(30,31)(33,38,39,42)(34,36,40,46)(35,45,37,48)(41,47,43,44)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=gi,d^2=ghi,e^2=h,f^2=hi,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, 8247339);