Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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 |
| prime radical | #24792 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,10,11,12)(15,21)(16,22)(18,24)(20,25)(23,27)(26,28);(1,2)(3,4)(6,9)(10,12)(13,14,17,19)(15,18,23,26)(16,20,22,25)(21,24,27,28);(2,4)(13,15,22,27,17,23,16,21)(14,18,25,28,19,26,20,24);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(14,19)(15,23)(20,25)(21,27);(2,4)(14,19)(15,23)(20,25)(21,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6490975);