Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119911 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator, prime radical | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
884 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 884 conjugacy classes of subgroups — too many to draw. Only its 126 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)(3,5)(4,6)(7,8)(9,10,14,18,13,17,21,24)(11,15,19,22,12,16,20,23);(1,3)(2,5)(4,7)(6,8)(9,11,13,12)(10,15,17,16)(14,20,21,19)(18,23,24,22);(5,8)(9,12,13,11)(10,16)(14,19,21,20)(15,17)(18,22)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,e^b=ei,g^b=gh,e^c=eh,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72303948, #72303950, #72303952, … |
| maximal normal subgroup of | #72303948, #72303950, #72303952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27109);