Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,12,15,20)(7,16,14,18);(2,4)(5,6)(7,17)(8,14)(9,20)(10,12)(11,15)(13,18)(16,19);(5,7)(6,13)(8,12)(9,16)(10,18)(11,14)(15,19)(17,20);(5,8,9,13,11,17,10,19)(6,14,20,18,15,7,12,16)Polycyclic presentation:
2^9|a^2=h,d^2=gh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250285, #72257167, #72257175 |
| maximal normal subgroup of | #72250285, #72257167, #72257175 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322805);