Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3990 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 3990 conjugacy classes of subgroups — too many to draw. Only its 354 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) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,7,10,8,9,11,12)(13,14)(15,21,23,28)(16,20)(17,19)(18,22,26,24)(25,27);(6,9)(7,11)(13,15)(14,18)(16,24)(17,23)(19,26)(20,28)(21,27)(22,25);(2,4);(14,19)(18,26)(20,27)(21,28)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72282667, #72282719, #72284371, … |
| maximal normal subgroup of | #72282667, #72282719, #72284371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92328);