Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5324 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,23,24)(19,21,20,22);(1,3,5,4)(2,6,8,7)(9,11,14,13)(10,15,12,16)(17,19)(18,22)(20,23)(21,24);(1,3,5,4)(2,7,8,6)(9,12,14,10)(11,16,13,15)(17,20,23,19)(18,21,24,22);(1,3,5,4)(2,7,8,6)(9,11,14,13)(10,16,12,15)(17,21,23,22)(18,20,24,19);(1,3,5,4)(2,7,8,6)(9,13,14,11)(10,16,12,15)(17,22,23,21)(18,20,24,19);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613457);