Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | C2×#3459 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2650 conjugacy classes of subgroups; only the 324 normal subgroups are shown.
This group has 2650 conjugacy classes of subgroups — too many to draw. Only its 324 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 — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(1,2)(3,5)(4,6)(7,8)(11,17)(14,20)(18,23)(21,24);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^9|a^2=g,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,f^b=fh,g^b=gh,e^c=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238180, #72238197, #72281840, … |
| maximal normal subgroup of | #72238180, #72238197, #72281840, … |
| maximal quotient of | #72278088, #72278185, #72278394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92111);