Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1889 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)(14,19)(15,23)(17,26)(20,25)(21,27)(22,28);(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,19,18,22,26,28)(15,20,23,16,21,25,27,24);(6,9)(10,11)(13,15,17,23,18,21,26,27)(14,20,19,16,22,25,28,24);(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,22)(17,25)(18,24)(19,27)(20,26)(23,28);(15,21)(16,24)(20,25)(23,27)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=g,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6371279);