Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 | #24439 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2010 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)(14,18)(15,19)(20,24)(21,26)(22,25)(27,28);(1,2)(3,4)(6,9)(10,12)(13,14,17,18)(15,19,22,25)(16,20,23,24)(21,26,27,28);(2,4)(13,15,17,22)(14,19,18,25)(16,21,23,27)(20,26,24,28);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16,17,23)(14,20,18,24)(15,21,22,27)(19,26,25,28);(2,4)(16,23)(20,24)(21,27)(26,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ehi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483490);