Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,15,16,11)(8,17)(9,19,20,13)(10,14)(12,18);(1,2)(3,4)(5,7)(6,11)(8,9)(10,16)(12,19)(13,17)(14,15)(18,20);(2,4)(5,8)(6,12,14,17)(7,9)(10,18)(11,13,15,19)(16,20);(5,9)(6,13)(7,8)(10,20)(11,17)(12,15)(14,19)(16,18)Polycyclic presentation:
2^9|a^2=h,c^2=e,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72234643, #72234649, #72234690, … |
| maximal normal subgroup of | #72234643, #72234649, #72234690, … |
| maximal quotient of | #72234647, #72240912, #72258609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267094);