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 | |
5189 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)(5,8)(6,7)(10,13)(14,16)(17,18)(19,23)(20,24)(21,22);(1,3,5,7)(2,6,8,4)(9,10,12,13)(11,14,15,16);(1,4)(2,7)(3,8)(5,6)(18,22)(19,24);(1,3,5,7)(2,6,8,4)(17,19,21,24)(18,23,22,20);(1,4)(2,3)(5,6)(7,8)(17,20,21,23)(18,24,22,19);(1,2)(3,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|b^2=g,d^2=h,e^2=hi,f^2=gi,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, 7602137);