Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9980 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | #36652 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1150 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1150 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,4,7,8)(5,6)(9,10,14,19,15,20,13,18)(11,21,24,16,25,32,23,31)(12,26,27,17,28,34,29,35)(22,30,37,40,39,33,38,36);(1,3)(2,4)(5,7)(6,8)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,33)(18,32)(19,21)(20,31)(26,36)(27,38)(28,39)(29,37)(30,35)(34,40);(1,4)(2,3)(5,8)(6,7)(9,12)(10,17)(11,22)(13,29)(14,27)(15,28)(16,30)(18,26)(19,34)(20,35)(21,36)(23,38)(24,37)(25,39)(31,33)(32,40)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,e^c=ei,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72303824 |
| maximal normal subgroup of | #72303824 |
| maximal quotient of | #72303965 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18521);