Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #160892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,9)(7,11)(10,12)(13,14)(15,18)(16,23,24,19)(17,20)(21,27,28,25)(22,26);(1,2,3,4)(13,15)(14,18)(16,21)(17,22)(19,25)(20,26)(23,27)(24,28);(5,6,8,10)(7,12,11,9)(13,16,17,24)(14,19,20,23)(15,21,22,28)(18,25,26,27);(15,22)(18,26)(21,28)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=h,f^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dhi,f^c=fi,g^c=ghi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95718);