Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#447 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14,18,22)(15,23,25,28)(16,21,17,20)(19,26,27,24);(1,2,3,4)(13,15,16,24,18,25,17,26)(14,19,20,23,22,27,21,28);(6,8)(15,24)(16,17)(19,23)(20,21)(25,26)(27,28);(10,12)(15,25)(19,27)(23,28)(24,26)Polycyclic presentation:
2^9|a^2=i,b^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264792, #72264812, #72264845 |
| maximal normal subgroup of | #72264792, #72264812, #72264845 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301511);