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 | |
5526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(5,7)(6,8)(9,10,14,11)(12,15,13,16)(17,18,21,20)(19,23,24,22);(1,3,6,5)(2,7,8,4)(17,19,21,24)(18,22,20,23);(1,4)(2,5)(3,8)(6,7)(10,11)(15,16)(17,18,21,20)(19,22,24,23);(1,3,6,5)(2,7,8,4)(9,11,14,10)(12,16,13,15);(1,4)(2,3)(5,8)(6,7)(9,12,14,13)(10,15,11,16)(17,20,21,18)(19,23,24,22);(1,5,6,3)(2,7,8,4)(9,13,14,12)(10,16,11,15)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7604477);