Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143142 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #2011 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18)(19,22)(20,29)(21,25)(23,26)(24,27)(28,31)(30,32);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30);(6,15)(9,16)(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30)Polycyclic presentation:
2^9|d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237263, #72265543, #72265553 |
| maximal normal subgroup of | #72237263, #72265543, #72265553 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50338);