Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,21)(20,24)(22,26)(23,25)(27,28);(1,2)(3,4)(6,9)(10,12)(13,15,17,22)(14,18,21,26)(16,23,24,28)(19,25,20,27);(2,4)(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(2,4);(5,7,8,11)(6,10,9,12)(15,22)(18,26)(23,28)(25,27)Polycyclic presentation:
2^9|b^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72256930, #72256947, #72256994, … |
| maximal normal subgroup of | #72256930, #72256947, #72256994, … |
| maximal quotient of | #72264145, #72264798, #72308824 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347116);