Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
918 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 918 conjugacy classes of subgroups — too many to draw. Only its 238 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(6,11)(7,12)(8,9)(14,18)(16,23)(19,24)(20,25)(22,27)(26,28);(5,6,8,11,10,12,9,7)(13,14,16,18,17,20,23,25)(15,19,22,24,21,26,27,28);(5,7,9,12,10,11,8,6)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(1,3,4,2)(6,12)(7,11)(15,21)(19,26)(22,27)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,d^c=dh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216602, #72224769, #72224772, … |
| maximal normal subgroup of | #72216602, #72224769, #72224772, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120085);