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 | #44932 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2186 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)(14,18)(15,21)(20,26)(22,24);(1,2,4,3)(7,12)(9,11)(15,22)(17,23)(19,27)(20,26)(21,24)(25,28);(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,18)(17,20)(21,25)(22,27)(23,26)(24,28);(6,10)(9,11)(13,15,16,21)(14,19,18,25)(17,24,23,22)(20,28,26,27);(1,3,4,2)(14,18)(19,25)(20,26)(27,28)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=i,e^2=f,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, 7441053);