Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130850 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8376 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | #61293 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
579 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 579 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,12,4,9,19,26,7,13,21,30,6,11,20,28)(3,14,18,29,17,8,23,24,16,10,22,27,15,25,31,32);(2,8)(3,15)(4,6)(5,20)(9,14)(10,11)(12,27)(13,25)(16,17)(18,22)(19,21)(24,26)(28,32)(29,30);(2,9,13,11)(3,16)(5,21)(8,14,25,10)(12,26,30,28)(15,17)(19,20)(24,29,32,27)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=hi,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,f^c=fgi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72231596, #72231704, #72231712, … |
| maximal normal subgroup of | #72231596, #72231704, #72231712, … |
| maximal quotient of | C2×#130852, #72250850, #72264850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38048);