Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153133 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13343 |
| commutator, Frattini, cosocle | #532 | C23 |
| prime radical | #31076 | C2 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
583 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 583 conjugacy classes of subgroups — too many to draw. Only its 50 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)(3,15)(4,21)(5,23)(6,25)(7,12)(8,14)(9,17)(10,11)(13,24)(16,19)(18,22)(20,29)(26,28)(27,32)(30,31);(1,3,8,20)(2,9,14,16)(4,22,13,28)(5,19,10,17)(6,26,7,18)(11,29,23,15)(12,30,25,27)(21,32,24,31);(1,4)(2,10)(3,16)(5,14)(6,23)(7,11)(8,13)(9,20)(12,24)(15,31)(17,28)(18,27)(19,22)(21,25)(26,30)(29,32)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fgh,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=gh,d^c=dg,e^c=ei,g^c=gi,h^c=hi,f^d=fhi,g^d=gi,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72209801, #72337616, #72337673, … |
| maximal normal subgroup of | #72337616, #72337673, #72340549 |
| maximal quotient of | #72209795, #72227653, #72227655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60330);