Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535648 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1677 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(17,18,21,23)(19,24,22,20);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,22,21,23,24,19);(10,13)(14,16)(17,19,20,18,21,22,24,23);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,24)(18,19,23,22);(11,15)(14,16)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=g,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313721, #72313734, #72313961 |
| maximal normal subgroup of | #72313721, #72313734, #72313961 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6468335);