Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251883 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1297 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 1297 conjugacy classes of subgroups — too many to draw. Only its 126 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)(4,14)(8,23)(9,22)(10,20)(13,30)(15,16)(18,25)(19,26)(21,31)(24,32)(28,29);(2,8,10,25)(3,11)(4,15)(7,18,20,23)(9,24,26,31)(12,27)(13,30)(14,16)(19,32,22,21)(28,29);(1,2)(3,7)(4,8)(5,10)(6,9)(11,22)(12,20)(13,21)(14,31)(15,25)(16,24)(17,26)(18,28)(19,27)(23,30)(29,32);(2,9)(3,12)(7,19)(8,24)(10,26)(11,27)(13,29)(18,32)(20,22)(21,23)(25,31)(28,30)Polycyclic presentation:
2^9|b^2=e,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^d=fh44 conjugacy classes — view the full character table.
| maximal subgroup of | #72234597, #72234627, #72237742, … |
| maximal normal subgroup of | #72234597, #72234627, #72237742, … |
| maximal quotient of | #72251435, #72251436, #72251457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159079);