Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125725 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | C22×D4 | D8 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 54 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:
(1,2,4,7)(3,5,8,6)(9,10,14,19)(11,20,23,33)(12,25,26,17)(13,16,29,34)(15,24,32,21)(18,28,35,27)(22,31,39,36)(30,37,40,38);(2,5)(3,8)(6,7)(9,11)(10,15)(12,24)(13,22)(14,23)(16,30)(17,33)(18,36)(19,32)(20,25)(21,26)(27,38)(28,37)(29,39)(31,35)(34,40);(10,16,19,34)(12,26)(13,28)(15,30,32,40)(17,18,25,35)(20,31,33,36)(21,24)(22,37)(27,29)(38,39)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=f,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=df,e^b=egi,d^c=dg,e^c=ehi,h^c=hi,e^d=ei41 conjugacy classes — view the full character table.
| maximal subgroup of | #72307558, #72307562, #72307579, … |
| maximal normal subgroup of | #72307558, #72307562, #72307579, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33124);