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 | |
3954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(4,11)(12,16)(18,20)(23,26)(24,28)(30,32)(33,34,38,41)(35,37,40,47)(36,43,46,39)(42,48,44,45);(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(19,25)(23,26)(27,31)(30,32)(33,35,38,40)(34,37,41,47)(36,44,46,42)(39,48,43,45);(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,36,38,46)(34,39,41,43)(35,42,40,44)(37,45,47,48);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,35)(34,37)(36,42)(38,40)(39,45)(41,47)(43,48)(44,46);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37)(34,40)(35,41)(36,45)(38,47)(39,44)(42,43)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,e^2=hi,f^2=hi,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, 8370938);