Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346482 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2367 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 2367 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,8)(10,16)(11,20)(13,14)(18,33)(19,25)(21,35)(22,27)(23,26)(28,31)(29,34)(30,39)(36,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,18)(13,25)(14,16)(15,19)(20,31)(21,30)(22,32)(23,29)(24,34)(26,33)(27,28)(35,40)(36,38)(37,39);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,21)(13,20)(14,27)(15,22)(16,28)(18,30)(19,32)(23,36)(24,37)(25,31)(26,35)(29,38)(33,40)(34,39);(9,12,15,24)(10,18,19,34)(11,21,22,37)(13,26,14,23)(16,29,25,33)(17,30,32,39)(20,35,27,36)(28,38,31,40)Polycyclic presentation:
2^9|c^2=i,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,e^d=eh53 conjugacy classes — view the full character table.
| maximal subgroup of | #72234939 |
| maximal normal subgroup of | #72234939 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253682);