Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(14,15)(17,18)(20,23)(21,22)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,14,18,21,19,22,17,15)(16,20,27,24,25,28,26,23);(2,4)(6,9)(10,12)(13,15,18,14,19,21,17,22)(16,24,27,28,25,23,26,20);(5,7)(6,10)(8,11)(9,12)(13,16,19,25)(14,20,22,28)(15,23,21,24)(17,26,18,27);(7,11)(10,12)(16,25)(20,28)(23,24)(26,27)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6459915);