Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| 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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2986 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)(3,4)(5,6)(7,12)(8,10)(9,11);(5,7,9,8)(6,10,11,12)(13,14)(15,19)(16,18)(17,21)(20,25)(22,26)(23,28)(24,27);(2,4)(14,18)(15,22)(16,25)(20,21)(23,24)(26,28);(5,8,9,7)(6,10,11,12)(13,15,17,24)(14,19,21,27)(16,22,25,23)(18,26,20,28);(14,18)(15,22)(16,25)(20,21)(23,24)(26,28)Polycyclic presentation:
2^9|b^2=f,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72256952, #72256956, #72257025, … |
| maximal normal subgroup of | #72256952, #72256956, #72257025, … |
| maximal quotient of | #72264981, #72265094, #72265143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483738);