Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14)(15,20)(16,21)(17,28)(18,22)(19,23)(24,25)(26,27);(1,2,3,4)(13,14)(15,24)(16,27)(17,22)(18,28)(19,23)(20,25)(21,26);(6,9)(10,12)(13,15,19,26)(14,20,23,27)(16,17,25,18)(21,28,24,22);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,17)(18,26)(19,25)(20,28)(22,27)(23,24);(13,17,19,18)(14,22,23,28)(15,25,26,16)(20,21,27,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72283676 |
| maximal normal subgroup of | #72283676 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357181);