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 |
| prime radical | #45422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,10)(9,12);(1,2,4,3)(7,12)(9,11)(13,14)(15,24)(16,20)(17,22)(18,27)(19,23)(21,26)(25,28);(5,7,8,12)(6,9,10,11)(13,15,19,16)(14,20,23,24)(17,26,18,25)(21,22,28,27);(6,10)(9,11)(13,16,19,15)(14,21,23,28)(17,26,18,25)(20,22,24,27);(1,3,4,2)(13,17,19,18)(14,22,23,27)(15,25,16,26)(20,21,24,28)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7441054);