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×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,19,23)(15,18,22,17)(16,21,27,20)(24,26,25,28);(1,2,3,4)(13,15,18,23,19,22,17,14)(16,20,26,25,27,21,28,24);(6,9)(10,12)(13,16,19,27)(14,20,23,21)(15,24,22,25)(17,26,18,28);(5,7,8,11)(6,10,9,12)(13,16,19,27)(14,21,23,20)(15,25,22,24)(17,26,18,28);(13,17,19,18)(14,22,23,15)(16,26,27,28)(20,25,21,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72283271 |
| maximal normal subgroup of | #72283271 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357333);