Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130844 |
| 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 | #61288 | 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 | |
572 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 572 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:
(1,2,6,13,4,17,3,14,8,10,24,29,7,12,16,30)(5,19,21,9,18,27,31,26,23,11,20,25,22,32,15,28);(1,3)(2,9)(4,6)(5,20)(7,24)(8,16)(10,25)(11,13)(12,26)(14,27)(15,22)(17,28)(18,31)(19,29)(21,23)(30,32);(1,4,8,7)(2,10)(3,6,16,24)(5,18,23,22)(9,25)(11,19)(12,17)(13,29)(14,30)(15,20,31,21)(26,28)(27,32)Polycyclic presentation:
2^9|a^2=f,c^2=gi,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,f^b=fgh,h^b=hi,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72231599, #72231712, #72231719, … |
| maximal normal subgroup of | #72231599, #72231712, #72231719, … |
| maximal quotient of | #72250426, #72257094, C2×#130847, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38043);