Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238038 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1075 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 1075 conjugacy classes of subgroups — too many to draw. Only its 174 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 — 4 generators on 32 points:
(2,7,10,12)(4,15)(8,23,16,20)(9,21,25,26)(13,30)(17,18)(22,24,29,32)(27,31);(2,8,10,16)(3,11)(7,20,12,23)(9,24,25,32)(13,27)(14,28)(21,22,26,29)(30,31);(1,2)(3,12)(4,16)(5,10)(6,9)(7,14)(8,17)(11,26)(13,22)(15,24)(18,32)(19,25)(20,27)(21,28)(23,30)(29,31);(2,9)(7,21)(8,24)(10,25)(12,26)(16,32)(20,22)(23,29)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,f^c=fh,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237865, #72237868, #72237976, … |
| maximal normal subgroup of | #72237865, #72237868, #72237976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145234);