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 | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1838 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 1838 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)(10,15)(13,26)(16,19)(17,23)(21,27)(25,28)(30,32)(37,38);(1,3)(2,5)(4,7)(6,8)(9,10,13,18,14,15,26,29)(11,19,21,34,22,16,27,31)(12,23,25,35,24,17,28,33)(20,30,37,39,36,32,38,40);(2,6)(5,8)(9,11,14,22)(10,16,15,19)(12,20,24,36)(13,27,26,21)(17,32,23,30)(18,34,29,31)(25,38,28,37)(33,39,35,40);(3,7)(5,8)(9,12)(10,17)(11,20)(13,28)(14,24)(15,23)(16,30)(18,35)(19,32)(21,38)(22,36)(25,26)(27,37)(29,33)(31,40)(34,39)Polycyclic presentation:
2^9|b^2=h,c^2=i,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, #72263056, #72267106 |
| maximal normal subgroup of | #72262956, #72263056, #72267106 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298911);