Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188252 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
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×#3484 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,14,16,11)(8,10)(9,13)(12,17)(15,19,20,18);(1,2)(3,4)(6,10)(8,17)(11,18)(12,13)(14,19)(15,20);(2,4)(5,7)(6,11)(8,15)(9,16)(10,18)(12,19)(13,14)(17,20);(6,10)(8,17)(11,18)(12,13)(14,19)(15,20)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214076, #72216142, #72216166, … |
| maximal normal subgroup of | #72214076, #72216142, #72216166, … |
| maximal quotient of | C2×#359639, #72239580, #72248564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266835);