Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1879 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)(13,14,19,24)(15,22,17,21)(16,26,27,25)(18,23,20,28);(1,2)(3,4)(6,9)(7,11)(13,15)(14,21)(16,23)(17,19)(18,26)(20,25)(22,24)(27,28);(2,4)(13,16,18,21,19,27,20,22)(14,17,28,25,24,15,23,26);(6,9)(10,12)(13,17,19,15)(14,22,24,21)(16,23,27,28)(18,25,20,26);(2,4)(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,23,24,28)(15,25,17,26)(16,21,27,22)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592613);