Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(14,18)(15,19)(16,24)(21,23)(22,27)(26,28);(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26);(1,2,4,3)(6,9)(10,12)(13,15,14,19,17,23,18,21)(16,22,20,26,24,28,25,27);(5,7,8,11)(6,10,9,12)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27);(1,3,4,2)(16,24)(20,25)(22,28)(26,27)Polycyclic presentation:
2^9|b^2=i,c^2=fgh,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724479);