Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184991 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3498 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,12,10,13,8,14)(7,15,16,19,17,11,18,20);(1,2,3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20);(2,4)(6,12)(8,9)(11,19)(13,14)(15,20)(16,18);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,15)(14,20)Polycyclic presentation:
2^9|a^2=h,b^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72227233, #72227268, #72227287, … |
| maximal normal subgroup of | #72227233, #72227268, #72227287, … |
| maximal quotient of | #72265377 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268830);