Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1519 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C2 | #7586 |
| prime radical | #3054 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
375 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 375 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,13,15,29,17,30,16,26)(4,18,19,9)(5,21,20,10)(6,12)(7,11)(8,14,24,22,25,31,28,32)(23,27);(1,3)(2,8)(4,14)(5,22)(6,15)(7,17)(9,13)(10,26)(11,25)(12,28)(16,23)(18,30)(19,31)(20,32)(21,29)(24,27);(1,4,6,20,7,19,23,5)(2,9,12,21,11,18,27,10)(3,14,15,32,17,31,16,22)(8,13,28,29,25,30,24,26)Polycyclic presentation:
2^9|a^2=e,c^2=gh,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219576, #72219579, #72219580, … |
| maximal normal subgroup of | #72219576, #72219579, #72219580, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31636);