Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #119915 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1440 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
876 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 876 conjugacy classes of subgroups — too many to draw. Only its 118 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:
(1,2,4,11)(3,9,14,25)(5,7,16,22)(6,10,17,26)(8,23,24,12)(13,19,27,30)(15,20,28,31)(18,21,29,32);(2,7)(3,12)(4,16)(8,20)(9,19)(10,21)(13,15)(14,27)(17,29)(23,28)(24,25)(30,31);(2,8,7,19)(3,13)(4,17)(9,21,20,10)(11,25,22,31)(12,15)(14,28)(16,29)(23,27)(24,32,30,26)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,e^b=ei,g^b=gh,e^c=ehi,g^c=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72233122, #72233126, #72233134, … |
| maximal normal subgroup of | #72233122, #72233126, #72233134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27111);