Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1856 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 1856 conjugacy classes of subgroups — too many to draw. Only its 206 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:
(1,2)(3,7,8,5)(4,6)(9,10)(11,19,22,15)(12,16)(13,25,26,17)(14,18)(20,27,34,29)(21,28)(23,36,37,30)(24,31)(32,35)(33,39)(38,40);(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,28)(18,19)(23,33)(24,34)(25,32)(26,35)(27,31)(30,39)(36,38)(37,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,24)(10,16,18,31)(11,20,22,34)(13,23,26,37)(15,27,19,29)(17,30,25,36)(21,33,35,40)(28,38,32,39);(9,13)(10,17)(11,21)(12,23)(14,26)(15,28)(16,30)(18,25)(19,32)(20,33)(22,35)(24,37)(27,38)(29,39)(31,36)(34,40)Polycyclic presentation:
2^9|a^2=g,c^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,g^c=gi,g^d=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72227298, #72227300, #72227723, … |
| maximal normal subgroup of | #72227298, #72227300, #72227723, … |
| maximal quotient of | #72266148, #72268081, #72268084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121026);