Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417579 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35676 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,25)(13,21)(14,19)(15,20)(17,26)(18,23)(22,34)(24,32)(27,36)(28,31)(29,35)(30,37)(33,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,22)(13,28)(14,23)(15,24)(17,33)(18,19)(20,32)(21,31)(25,34)(26,38)(27,39)(29,37)(30,35)(36,40);(9,12,15,27)(10,17,20,35)(11,22,24,39)(13,29,14,26)(16,30,32,33)(18,34,31,40)(19,36,21,25)(23,38,28,37);(3,8)(5,7)(9,11)(10,18)(12,22)(13,23)(14,28)(15,24)(16,19)(17,34)(20,31)(21,32)(25,33)(26,37)(27,39)(29,38)(30,36)(35,40)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318719, #72318775, #72318935, … |
| maximal normal subgroup of | #72318719, #72318775, #72318935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325132);