Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137387 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33669 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
372 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,3,7,11)(4,10,13,16,12,8,14,15)(17,18,19,25,21,28,20,24,23,30,32,26,22,29,31,27);(1,3,5,11,6,2,7,9)(4,10,14,15,12,8,13,16)(18,24,30,27)(19,20)(21,22)(25,29,26,28)(31,32);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,16)(11,15)(18,25)(19,31)(20,32)(21,22)(24,29)(26,30)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,d^2=g,e^2=gi,f^2=hi,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 | #72285149, #72285170, #72285367 |
| maximal normal subgroup of | #72285149, #72285170, #72285367 |
| maximal quotient of | #72343330, #72343333, #72343335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44675);