Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149557 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11426 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | #379 | C23 |
| prime radical | C2×#534 | C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,5,10)(3,11,13,25,14,7,17,20)(4,8,16,22)(6,9,18,24)(12,21,27,31,26,19,29,30)(15,23,28,32);(1,3)(2,7)(4,12)(5,17)(6,14)(8,21)(9,11)(10,20)(13,18)(15,26)(16,29)(19,23)(22,31)(24,25)(27,28)(30,32);(1,4,6,15)(2,8,9,23)(3,12,14,26)(5,16,18,28)(7,19,11,21)(10,22,24,32)(13,27,17,29)(20,30,25,31)Polycyclic presentation:
2^9|a^2=e,c^2=i,e^2=g,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ch,e^b=ef,g^b=gi,d^c=di,e^d=eh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72338693, #72338696, #72338699, … |
| maximal normal subgroup of | #72338693, #72338696, #72338699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56754);