Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2729 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)(7,11)(8,10)(9,12)(13,14,20,23)(15,18,22,19)(16,26,28,24)(17,27,21,25);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,21)(17,28)(19,23)(20,22)(24,27)(25,26);(13,16,20,28)(14,21,23,17)(15,24,22,26)(18,25,19,27);(2,4)(13,17,20,21)(14,16,23,28)(15,25,22,27)(18,26,19,24);(6,10)(9,11)(13,18,20,19)(14,22,23,15)(16,25,28,27)(17,24,21,26)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,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=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421871);