Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,14,17,20)(15,22,24,19)(16,26,18,21)(23,27,28,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,25)(17,24)(18,27)(20,22)(21,23)(26,28);(2,4)(6,9)(10,11)(13,14)(15,23)(16,21)(17,20)(18,26)(19,27)(22,25)(24,28);(5,7,8,12)(6,10,9,11)(14,20)(19,22)(21,26)(23,28);(13,16)(14,21)(15,24)(17,18)(20,26)(23,28)Polycyclic presentation:
2^9|a^2=f,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72308154, #72308437, #72308867 |
| maximal normal subgroup of | #72308154, #72308437, #72308867 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578032);