Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6)(7,8)(10,16)(11,19)(12,14)(13,18)(17,24)(20,23);(1,2)(3,4)(5,6)(7,8)(9,10,12,18,15,13,14,16)(11,20,22,23,19,17,21,24);(6,8)(9,11)(10,17)(12,22)(13,20)(14,21)(15,19)(16,23)(18,24);(2,4)(6,8)(9,12,15,14)(10,18,13,16)(11,21,19,22)(17,23,20,24);(9,13,14,18,15,10,12,16)(11,17,21,23,19,20,22,24)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=f,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^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281843, #72282960, #72283139, … |
| maximal normal subgroup of | #72281843, #72282960, #72283139, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426186);