Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128057 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
866 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 866 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6,8,7)(9,10)(11,19,21,15)(12,16)(13,18)(14,17)(20,24,22,23);(1,2)(3,7,4,6)(5,8)(9,11,13,22,14,21,12,20)(10,15,18,24,17,19,16,23);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gi,d^2=i,f^2=g,b^a=bd,d^a=dg,f^a=fhi,c^b=ch,d^b=dh,e^b=egi,h^b=hi,f^c=fi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72231143, #72231177, #72231188, … |
| maximal normal subgroup of | #72231143, #72231177, #72231188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35394);