Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137387 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 425 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(1,2,5,9,6,10,4,8)(3,11,13,7,14,15,12,16)(17,18,21,25,31,29,20,28,23,27,32,19,22,26,30,24);(1,3)(2,7)(4,13)(5,12)(6,14)(8,15)(9,11)(10,16)(17,19,22,24,23,25,31,28)(18,20,29,21,27,30,26,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,15)(10,16)(17,18)(19,20)(21,24)(22,29)(23,27)(25,30)(26,31)(28,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfhi,d^2=g,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eghi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eg,f^c=fh,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265643, #72308481, #72308493 |
| maximal normal subgroup of | #72265643, #72308481, #72308493 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44640);