Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129767 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | #60688 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
699 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 699 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(2,6,7,8)(10,15,16,23)(11,19)(12,13)(17,20,18,24)(21,22);(1,2)(3,6)(4,8)(5,7)(9,10)(11,20)(12,18)(13,17)(14,16)(15,22)(19,23)(21,24);(1,3,5,4)(2,6,7,8)(11,21)(15,23)(19,22)(20,24)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,h^b=hi,d^c=di56 conjugacy classes — view the full character table.
| maximal subgroup of | #72264995, #72265082, #72265087, … |
| maximal normal subgroup of | #72264995, #72265082, #72265087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37127);