Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #357285 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C22×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13827 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1597 conjugacy classes of subgroups; only the 312 normal subgroups are shown.
This group has 1597 conjugacy classes of subgroups — too many to draw. Only its 312 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 — 4 generators on 32 points:
(1,2,5,10,6,9,19,24)(3,11,14,25,13,7,18,21)(4,8,16,23,17,22,29,31)(12,20,27,30,15,26,28,32);(1,3)(2,7)(4,15,17,12)(5,18)(6,13)(8,20,22,26)(9,11)(10,25)(14,19)(16,27,29,28)(21,24)(23,32,31,30);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,23)(11,26)(13,15)(14,27)(18,28)(19,29)(21,30)(24,31)(25,32);(2,9)(5,19)(8,22)(14,18)(16,29)(21,25)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cef,d^b=de,f^b=fi,g^b=gi,h^c=hi,g^d=gi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72235132, #72235140, #72235225, … |
| maximal normal subgroup of | #72235132, #72235140, #72235225, … |
| maximal quotient of | #72235160 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264482);