Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232181 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1164 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1164 conjugacy classes of subgroups — too many to draw. Only its 190 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)(3,6)(4,5)(7,8)(9,10,14,19,13,20,15,18)(11,21,22,24,12,17,23,16);(3,7)(6,8)(11,22)(12,23)(14,15)(16,17)(18,19)(21,24);(2,5)(6,8)(9,11,13,12)(10,16,20,24)(14,22,15,23)(17,19,21,18);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,13)(14,23)(15,22)(16,19)(18,24)(20,21)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=fg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338919, #72338922, #72338940, … |
| maximal normal subgroup of | #72338919, #72338922, #72338940, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298851);