Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3264 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(14,18)(15,22)(16,19)(20,23)(21,26)(24,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,14,19,26,17,21,16,18)(15,23,27,25,24,20,22,28);(7,12)(9,10)(13,15,19,27,17,24,16,22)(14,20,26,28,21,23,18,25);(5,7)(6,10)(8,12)(9,11)(15,24)(20,23)(22,27)(25,28);(7,12)(9,10)(14,21)(18,26)(20,23)(25,28)Polycyclic presentation:
2^9|b^2=fh,c^2=gi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6514376);