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 | #2535 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3948 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)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32)(33,34,38,42)(35,37,41,39)(36,44,47,40)(43,46,45,48);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(18,24)(19,20)(26,31)(27,28)(33,35,38,41)(34,39,42,37)(36,45,47,43)(40,48,44,46);(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32)(33,36,38,47)(34,40,42,44)(35,43,41,45)(37,48,39,46);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)(33,37,38,39)(34,41,42,35)(36,46,47,48)(40,43,44,45)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8344033);