Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3486 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)(7,11)(10,12)(13,14)(15,23)(16,22)(17,21)(18,20)(19,25)(24,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,25)(17,27)(18,26)(20,24)(21,23)(22,28);(2,4)(6,9)(10,12)(13,14)(15,24)(16,21)(17,22)(18,20)(19,26)(23,27)(25,28);(5,7,8,11)(6,10,9,12)(14,20)(19,24)(21,22)(23,28);(13,16,18,17)(14,21,20,22)(15,25,26,27)(19,23,24,28)Polycyclic presentation:
2^9|d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281914, #72284213, #72285185 |
| maximal normal subgroup of | #72281914, #72284213, #72285185 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342218);