Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152678 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13082 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #23908 | C2 |
| LCS | G ▷ C22×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
630 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 630 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,7,13,21)(4,15,16,8)(5,11,19,9)(6,10)(12,20,26,31)(14,22,18,24)(17,25)(23,30)(27,29,28,32);(1,3,6,14)(2,7,11,21,10,22,9,24)(4,12,16,27)(5,18,19,13)(8,20,23,32,15,29,25,31)(17,26,30,28);(1,4)(2,8)(3,12)(5,17)(6,16)(7,20)(9,25)(10,15)(11,23)(13,28)(14,27)(18,26)(19,30)(21,32)(22,29)(24,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dgi,e^b=ei,f^b=fgi,g^b=gh,d^c=di,f^c=fi,f^d=fhi,g^d=gi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72231244, C2×#152678, #72279489, … |
| maximal normal subgroup of | #72231244, C2×#152678, #72279489, … |
| maximal quotient of | C2×#152678, #72279394, #72279404 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59874);