Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #160892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,12,10,7,8,11)(13,14)(15,19)(16,18)(17,21)(20,24)(22,26)(23,25)(27,28);(5,7,10,6)(8,11,9,12)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(2,4)(13,15)(14,19)(16,23)(17,22)(18,25)(20,27)(21,26)(24,28);(15,22)(19,26)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=hi,e^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=ghi,d^c=dh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92389);