Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94441 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7307 |
| commutator | C2×C42 | C42 |
| Frattini, prime radical, cosocle | C2×#535 | C22 |
| LCS | G ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#535 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
489 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 489 conjugacy classes of subgroups — too many to draw. Only its 44 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 — 2 generators on 32 points:
(1,2)(3,13,15,27)(4,18,19,26)(5,11,6,10)(7,12)(8,14,22,29)(9,20)(16,32,17,28)(21,25)(23,31,24,30);(1,3,5,15,7,17,6,16)(2,8,10,23,12,24,11,22)(4,14,20,29,19,30,21,31)(9,13,18,27,25,28,26,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,e^2=i,f^2=i,b^a=bc,c^a=cf,e^a=eghi,f^a=fh,h^a=hi,c^b=cg,d^b=dfh,g^b=gi,d^c=dhi,e^c=ei,e^d=ei,f^d=fi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72213651, #72213687, #72213697, … |
| maximal normal subgroup of | #72213651, #72213687, #72213697, … |
| maximal quotient of | #72338270 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1638);