Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2184 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)(7,11)(8,10)(9,12)(13,14)(15,20)(16,17)(18,21)(19,24)(22,23)(25,28)(26,27);(5,7,9,8)(6,10,12,11)(13,15,17,24)(14,19,16,20)(18,23,27,25)(21,28,26,22);(2,4)(5,7,9,8)(6,11,12,10)(13,16,17,14)(15,22,24,28)(18,21,27,26)(19,23,20,25);(5,8,9,7)(6,10,12,11)(13,15,17,24)(14,20,16,19)(18,23,27,25)(21,22,26,28);(14,16)(15,23)(18,27)(19,22)(20,28)(24,25)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257018, #72257035, #72257071, … |
| maximal normal subgroup of | #72257018, #72257035, #72257071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484358);