Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60750 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,26)(27,28);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,23)(17,24)(19,27)(20,22)(21,26)(25,28);(7,12)(9,11)(14,19)(15,23)(17,25)(18,26)(20,21)(22,27);(6,10)(9,11)(14,20)(18,22)(19,21)(26,27);(14,20)(18,22)(19,21)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=cf,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72284972, #72285050, #72285944 |
| maximal normal subgroup of | #72284972, #72285050, #72285944 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281504);