Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
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 | C23×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2404 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)(5,6)(7,12)(8,10)(9,11)(13,14,17,20)(15,19,22,27)(16,18,24,21)(23,25,28,26);(5,7,9,8)(6,10,11,12);(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(1,3,4,2)(5,8,9,7)(6,10,11,12)(13,15)(14,19)(16,23)(17,22)(18,25)(20,27)(21,26)(24,28);(14,20)(15,22)(18,21)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=fhi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,g^c=gi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72310629, #72310637, #72310655, … |
| maximal normal subgroup of | #72310629, #72310637, #72310655, … |
| maximal quotient of | #72312905, #72313336 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280416);