Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #114125 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#3151 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1378 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 1378 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10)(11,17,20,14)(12,16)(13,15)(18,23,19,22)(21,24);(2,6)(3,4)(7,8)(9,11)(10,14)(12,18)(13,20)(15,17)(16,22)(19,21)(23,24);(2,6)(3,4)(7,8)(11,18)(12,21)(14,22)(16,24)(17,23)(19,20)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214129, #72214488, #72214607, … |
| maximal normal subgroup of | #72214129, #72214488, #72214607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 21325);