Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417901 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
942 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 942 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,15,17,11)(8,14)(9,12)(10,13)(16,19,18,20);(1,2,3,4)(5,7,8,16,10,17,9,18)(6,11,14,20,13,15,12,19);(2,4)(7,16,17,18)(11,19,15,20);(6,12,13,14)(11,20,15,19)Polycyclic presentation:
2^9|a^2=f,b^2=efg,c^2=f,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72338513, #72338521, #72338522, … |
| maximal normal subgroup of | #72338513, #72338521, #72338522, … |
| maximal quotient of | #72235146, #72235150 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346575);