Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(6,7)(10,14)(12,17)(15,20)(16,21)(19,22)(23,24);(1,2)(3,4)(6,7)(9,10,12,14,13,16,17,21)(11,15,19,20,18,23,22,24);(5,6,8,7)(9,11)(10,15)(12,19)(13,18)(14,20)(16,23)(17,22)(21,24);(2,4)(5,7,8,6)(10,16)(11,18)(14,21)(19,22);(9,10,17,21,13,16,12,14)(11,15,22,24,18,23,19,20)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283116, #72283132, #72283139, … |
| maximal normal subgroup of | #72283116, #72283132, #72283139, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470449);