Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147767 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10991 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1806 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 1806 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,13,17,24)(4,9)(5,11,6,10)(7,12)(8,15)(14,25,29,32)(16,23)(18,31)(19,26,21,28)(20,27)(22,30);(1,3)(2,8)(4,18)(5,16)(6,15)(7,17)(9,25)(10,24)(11,13)(12,23)(14,19)(20,30)(21,29)(22,26)(27,32)(28,31);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,26)(11,28)(12,27)(13,25)(15,18)(16,30)(17,29)(23,31)(24,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,f^c=fi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213504, #72238712, #72238714, … |
| maximal normal subgroup of | #72213504, #72238712, #72238714, … |
| maximal quotient of | #72251666, C2×#147767 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54963);