Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201918 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3003 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,14,17)(11,13,16,23)(12,19,22,18)(15,20,24,21);(1,2)(3,5)(4,6)(7,8)(9,11,13,17,14,16,23,10)(12,18,20,24,22,19,21,15);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,18)(13,20)(14,22)(16,19)(17,24)(21,23);(9,13,14,23)(10,16,17,11)(12,20,22,21)(15,19,24,18)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308356, C2×#201920 |
| maximal normal subgroup of | #72308356, C2×#201920 |
| maximal quotient of | #72308368, #72308382, C2×#201920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109116);