Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196338 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#534 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,15,16,11)(9,12)(13,17,20,19)(14,18);(2,4)(5,7)(6,10)(8,13)(9,14)(11,19)(12,18)(15,17)(16,20);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,20)(18,19);(2,4)Polycyclic presentation:
2^9|a^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213862, #72218840, #72218980, … |
| maximal normal subgroup of | #72213862, #72218840, #72218980, … |
| maximal quotient of | #72213659, #72218846, #72219038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266638);