Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150615 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11808 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #379 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
452 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,15,17,8)(5,18)(6,10)(7,13,23,19)(9,21)(12,26,29,32)(14,27)(16,24,20,25)(22,28)(30,31);(1,3)(2,7)(4,12)(5,19,21,13)(6,14)(8,22)(9,11,18,27)(10,23)(15,31)(16,30,20,28)(17,29)(24,32,25,26);(1,4,6,17)(2,8,10,15)(3,12,14,29)(5,20,21,16)(7,22,23,31)(9,25,18,24)(11,26,27,32)(13,30,19,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=gh,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efgh,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=eh,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72283476, #72283548, #72284056, … |
| maximal normal subgroup of | #72283476, #72283548, #72284056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57812);