Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4356 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)(14,19)(15,23)(16,18)(20,27)(21,22)(24,28);(2,4)(13,14)(15,24)(16,22)(17,21)(18,19)(20,26)(23,27)(25,28);(5,7,9,8)(6,10,11,12)(13,15,17,23)(14,20,21,28)(16,26,18,25)(19,24,22,27);(5,8,9,7)(6,10,11,12)(14,19)(16,18)(20,24)(21,22)(25,26)(27,28);(14,21)(19,22)(20,28)(24,27)Polycyclic presentation:
2^9|c^2=g,d^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72234968, #72257755, #72257766, … |
| maximal normal subgroup of | #72234968, #72257755, #72257766, … |
| maximal quotient of | #72237004, #72257801, #72316134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359023);