Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10557 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | #60734 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1142 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1142 conjugacy classes of subgroups — too many to draw. Only its 142 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)(11,20,24,35)(12,25,29,17)(13,18)(14,30)(15,19)(16,22,32,23)(21,33)(26,31)(27,36,28,34)(37,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,23)(14,22)(15,24)(17,33)(18,35)(19,32)(20,30)(21,28)(25,37)(27,39)(29,38)(31,34)(36,40);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,21)(13,28)(14,27)(15,29)(16,31)(18,36)(19,25)(20,37)(22,26)(23,38)(24,39)(30,34)(32,40)(33,35)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^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=eh,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72313677, #72313680, #72313733 |
| maximal normal subgroup of | #72313677, #72313680, #72313733 |
| maximal quotient of | #72314179, #72314219 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18624);