Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1198 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 1198 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,14)(11,17)(12,19)(15,24)(16,23)(18,21)(20,22);(9,10)(11,18)(12,21)(13,16)(14,17)(15,19)(20,24)(22,23);(2,4);(9,11)(10,15)(12,22)(13,20)(14,23)(16,21)(17,19)Polycyclic presentation:
2^9|a^2=g,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#2177, C2×#497432, #72319320, … |
| maximal normal subgroup of | D4×#2177, C2×#497432, #72319320, … |
| maximal quotient of | #72225566, #72226890, C2×#497432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404628);