Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188062 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#560 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2944 conjugacy classes of subgroups; only the 310 normal subgroups are shown.
This group has 2944 conjugacy classes of subgroups — too many to draw. Only its 310 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:
(1,2,4,10)(3,9,13,24)(5,7,15,20)(6,8,16,22)(11,23,25,32)(12,19,26,30)(14,21,27,31)(17,18,28,29);(2,7)(3,11)(4,15)(8,18)(9,21)(12,14)(13,27)(16,28)(19,23)(24,32)(25,26)(30,31);(2,8)(3,12)(4,16)(7,18)(9,23)(11,14)(13,25)(15,28)(19,21)(24,30)(26,27)(31,32);(1,3,5,12)(2,9,7,19)(4,13,15,26)(6,14,17,11)(8,21,18,23)(10,24,20,30)(16,27,28,25)(22,31,29,32)Polycyclic presentation:
2^9|a^2=g,d^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,g^b=gh,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234501, #72234504, #72234506, … |
| maximal normal subgroup of | #72234501, #72234504, #72234506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95261);