Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62253 |
| 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 | #60734 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1980 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)(9,10,14,18,15,19,13,28)(11,16,22,30,23,31,21,37)(12,24,27,32,25,38,26,17)(20,33,36,39,34,40,35,29);(1,3)(2,5)(4,8)(6,7)(9,11,14,22,15,23,13,21)(10,16,18,30,19,31,28,37)(12,20,27,36,25,34,26,35)(17,29,24,33,32,39,38,40);(9,12)(10,17,19,32)(11,20)(13,26)(14,27)(15,25)(16,29,31,39)(18,24,28,38)(21,35)(22,36)(23,34)(30,33,37,40);(3,8)(5,7)(12,25)(17,32)(20,34)(24,38)(26,27)(29,39)(33,40)(35,36);(2,6)(5,7)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,d^b=df,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72265375, #72308656, #72308688 |
| maximal normal subgroup of | #72265375, #72308656, #72308688 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6303358);