Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3558 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,25)(13,18)(14,19)(15,20)(17,28)(21,34)(22,31)(23,32)(24,33)(26,30)(27,35)(29,36)(37,39)(38,40);(2,5)(6,8)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,34)(18,31)(19,32)(20,33)(21,28)(25,30)(27,37)(29,38)(35,39)(36,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,21,24,37)(13,27)(14,28)(15,29)(16,30,33,40)(18,25)(19,36)(20,35)(22,26,23,38)(31,39,32,34);(3,7)(6,8)(9,13,15,14)(10,18,20,19)(11,22,24,23)(12,27,29,28)(16,31,33,32)(17,25,35,36)(21,26,37,38)(30,39,40,34)Polycyclic presentation:
2^9|c^2=h,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285739, #72285809, #72286066, … |
| maximal normal subgroup of | #72285739, #72285809, #72286066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289612);