Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34936 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(9,10)(11,19)(12,21)(13,23)(14,17)(15,20)(16,22)(18,24);(1,2)(3,4)(5,6)(7,8);(6,8)(9,11,14,12)(10,15,17,16)(13,22,24,20)(18,21,23,19);(2,4)(9,12,14,11)(10,16,17,15)(13,20,24,22)(18,19,23,21);(10,17)(15,16)(18,23)(19,21)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283257, #72283296, #72284539, … |
| maximal normal subgroup of | #72283257, #72283296, #72284539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408141);