Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60749 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,19,15,20,13,17)(11,16,24,28,21,30,23,31)(12,18,27,33,25,34,26,32)(22,29,37,38,35,40,36,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,24,15,21,13,23)(10,16,19,28,20,30,17,31)(12,22,27,37,25,35,26,36)(18,29,33,38,34,40,32,39);(10,17,20,19)(11,21)(12,25)(16,28,30,31)(18,33,34,32)(23,24)(26,27)(29,39,40,38);(3,8)(5,7)(9,12)(10,18)(11,22)(13,26)(14,27)(15,25)(16,29)(17,32)(19,33)(20,34)(21,35)(23,36)(24,37)(28,38)(30,40)(31,39);(2,6)(5,7)(12,25)(18,34)(22,35)(26,27)(29,40)(32,33)(36,37)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72308458, #72308688, #72308766, … |
| maximal normal subgroup of | #72308458, #72308688, #72308766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6303366);