Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,15,11,7,10,14)(8,16,19,17,20,12,18,13);(5,7)(6,11)(8,17)(9,14)(10,15)(12,19)(13,20)(16,18);(2,4)(5,8)(6,12)(7,16)(9,18)(10,19)(11,20)(13,14)(15,17);(5,8)(6,13)(7,17)(9,19)(10,18)(11,20)(12,14)(15,16)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239611, #72265192, #72265640, … |
| maximal normal subgroup of | #72239611, #72265192, #72265640, … |
| maximal quotient of | #72239825, #72239834, #72286769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266883);