Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233559 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
2157 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 2157 conjugacy classes of subgroups — too many to draw. Only its 190 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)(3,7)(4,8)(5,9)(6,11)(10,17)(12,26)(13,21)(14,19)(15,25)(16,24)(18,29)(20,30)(22,28)(23,31)(27,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,22)(9,21)(10,20)(11,19)(15,29)(16,28)(17,30)(18,23)(24,26)(25,32)(27,31);(1,4)(2,8)(3,12)(5,16)(6,15)(7,18)(9,24)(10,23)(11,25)(13,28)(14,29)(17,31)(19,26)(20,22)(21,32)(27,30);(2,9)(3,13)(4,15)(8,23)(10,11)(12,27)(14,30)(16,31)(18,26)(22,32)(24,25)(28,29)Polycyclic presentation:
2^9|b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,d^c=di,e^c=eh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72251101, #72251150, #72251153, … |
| maximal normal subgroup of | #72251101, #72251150, #72251153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140755);