Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213744 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35676 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1808 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1808 conjugacy classes of subgroups — too many to draw. Only its 234 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,20,15)(12,16,24,32)(13,26,27,17)(21,31,23,29)(22,30,36,35)(25,34,28,33)(37,39,38,40);(1,3)(2,5)(4,7)(6,8)(9,11,13,22,14,20,27,36)(10,15,17,30,18,19,26,35)(12,23,25,38,24,21,28,37)(16,31,33,40,32,29,34,39);(2,6)(5,8)(11,20)(13,27)(15,19)(17,26)(25,28)(33,34)(37,38)(39,40);(3,7)(5,8)(9,12)(10,16)(11,21)(13,28)(14,24)(15,29)(17,34)(18,32)(19,31)(20,23)(22,38)(25,27)(26,33)(30,40)(35,39)(36,37)Polycyclic presentation:
2^9|a^2=i,b^2=h,h^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72262956, #72263057, #72267104, … |
| maximal normal subgroup of | #72262956, #72263057, #72267104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298912);