Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #113288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #34632 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1797 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1797 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,19,22,32)(12,23,25,16)(13,17)(14,18)(15,21,29,26)(20,34)(24,37,39,31)(27,33)(28,38)(30,35)(36,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,26)(14,22)(16,30)(17,32)(18,29)(19,33)(21,27)(23,28)(24,38)(25,36)(31,34)(35,39)(37,40);(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,31)(18,23)(19,34)(21,35)(22,36)(26,38)(27,39)(29,30)(32,40)(33,37)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,g^c=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72235168, #72235200, #72255873, … |
| maximal normal subgroup of | #72235168, #72235200, #72255873, … |
| maximal quotient of | #72235923 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20484);