Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135848 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#41 | C4⋊C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
291 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10,14,19,15,21,13,20)(11,22,24,17,25,37,31,38)(12,26,29,18,30,39,28,36)(16,32,34,23,35,27,33,40);(1,3,5,6,4,7,8,2)(9,11,15,25)(10,16)(12,27,30,32)(13,24,14,31)(17,36)(18,38)(19,33)(20,34)(21,35)(22,26)(23,28,40,29)(37,39);(2,6)(3,7)(9,11)(10,17)(12,23)(13,31)(14,24)(15,25)(16,18)(19,37)(20,22)(21,38)(26,33)(27,29)(28,32)(30,40)(34,39)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,d^b=dg,f^b=fi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319890, #72319904, #72319956, … |
| maximal normal subgroup of | #72319890, #72319904, #72319956, … |
| maximal quotient of | C2×#135848 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43044);