Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150723 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11839 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #379 | C23 |
| prime radical | #23988 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #2309 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 52 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)(7,13,22,30)(9,20)(12,27,29,26)(14,24)(16,23,19,25)(21,28)(31,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,21)(9,24)(10,22)(11,18)(15,31)(16,28)(17,29)(19,32)(20,30)(23,27)(25,26);(1,4)(2,8)(3,12)(5,19)(6,17)(7,21)(9,25)(10,15)(11,26)(13,28)(14,29)(16,20)(18,23)(22,31)(24,27)(30,32)Polycyclic presentation:
2^9|a^2=e,d^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=efg,h^b=hi,d^c=dhi,e^c=ei,e^d=eh,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72338699, #72338725, #72338735, … |
| maximal normal subgroup of | #72338699, #72338725, #72338735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57919);