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×C4 |
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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(13,14,17,19)(15,23,24,28)(16,22,18,21)(20,26,27,25);(1,3,4,2)(5,6,8,9)(7,10,11,12)(14,19)(15,24)(21,22)(25,26);(6,9)(10,12)(13,15)(14,20)(16,25)(17,24)(18,26)(19,27)(21,23)(22,28);(5,7,8,11)(6,10,9,12)(14,19)(20,27)(21,22)(23,28);(13,16,17,18)(14,21,19,22)(15,25,24,26)(20,23,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=fi,e^2=i,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 | #72283507, #72284228, #72284386 |
| maximal normal subgroup of | #72283507, #72284228, #72284386 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283050);