Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C8 | C23 |
| socle | C22 | C2×#327 |
| prime radical | C24 | #102 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
347 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,8,5,7,12,15)(3,9,10,16,11,6,13,14)(17,18)(19,29,32,46)(20,25)(21,35)(22,27)(23,28)(24,31,40,47)(26,34)(30,45,33,43)(36,41,48,39)(37,44)(38,42);(1,3,5,11)(2,6,7,9)(4,10,12,13)(8,16,15,14)(17,19,23,32)(18,24,28,40)(20,30,22,33)(21,36,38,48)(25,39,27,41)(26,43,44,45)(29,35,46,42)(31,34,47,37);(17,20,23,22)(18,25,28,27)(19,30,32,33)(21,34,38,37)(24,39,40,41)(26,42,44,35)(29,45,46,43)(31,36,47,48)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=i,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29857);