Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 | C2×#3321 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1890 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)(7,10)(8,9)(11,12)(13,14,20,24)(15,18,23,19)(16,22,27,28)(17,25,26,21);(1,2)(3,4)(6,9)(10,12)(13,15)(14,18)(16,21)(17,22)(19,24)(20,23)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16,19,17,20,27,18,26)(14,21,23,28,24,25,15,22);(2,4)(13,17,19,27,20,26,18,16)(14,22,23,21,24,28,15,25);(6,9)(10,12)(13,18,20,19)(14,23,24,15)(16,26,27,17)(21,28,25,22)Polycyclic presentation:
2^9|a^2=i,c^2=fgi,d^2=gi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468346);