Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190116 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2606 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 2606 conjugacy classes of subgroups — too many to draw. Only its 254 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 — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15,16,20)(11,18)(13,21)(14,22,17,23)(19,24);(12,21)(15,20)(18,24)(22,23)Polycyclic presentation:
2^9|a^2=g,c^2=e,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219311, #72219325, #72219340, … |
| maximal normal subgroup of | #72219311, #72219325, #72219340, … |
| maximal quotient of | #72236988, #72255630, #72255653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97312);