Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161262 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
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 | C2×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3196 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)(5,6,8,10,9,11,7,12)(13,14)(15,21)(16,24)(17,19)(18,22)(20,26)(23,28)(25,27);(6,10)(7,8)(11,12)(13,15,17,23)(14,18,19,27)(16,25,26,22)(20,28,24,21);(2,4)(13,14)(15,18)(16,20)(17,19)(21,22)(23,27)(24,26)(25,28);(6,11)(10,12)(14,19)(18,27)(20,24)(21,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=ghi,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92412);