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 |
| prime radical | #2456 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3905 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)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(20,28)(24,29)(27,32)(30,31)(33,34,38,42)(35,39,45,48)(36,37,41,40)(43,44,47,46);(2,7)(4,14)(8,11)(10,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,30)(31,32)(33,35,38,45)(34,39,42,48)(36,43,41,47)(37,44,40,46);(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,36,38,41)(34,40,42,37)(35,43,45,47)(39,46,48,44);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,13,15,9)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,35)(34,39)(36,43)(37,44)(38,45)(40,46)(41,47)(42,48);(1,4)(2,9)(3,10)(5,8)(6,14)(7,13)(11,15)(12,16)(17,20,22,28)(18,24,26,29)(19,27,25,32)(21,30,23,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5)(2,10)(3,9)(4,8)(6,15)(7,16)(11,14)(12,13)(17,21,22,23)(18,25,26,19)(20,30,28,31)(24,32,29,27)(33,37,38,40)(34,41,42,36)(35,44,45,46)(39,47,48,43)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8413455);