Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2211 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)(7,9)(8,10)(11,12)(13,14,18,16,19,23,20,24)(15,17,22,28,26,27,25,21);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,21)(16,27)(17,24)(18,25)(19,26)(20,22)(23,28);(2,4)(6,10)(9,12)(13,16,18,23,19,24,20,14)(15,21,22,17,26,28,25,27);(2,4)(13,17,18,28,19,27,20,21)(14,22,16,26,23,25,24,15);(13,18,19,20)(14,16,23,24)(15,25,26,22)(17,21,27,28)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=f,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348984);