Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198162 |
| 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 | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2704 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)(3,5,7,8)(4,6)(9,10,13,16,14,18,12,17)(11,19,21,23,22,24,20,15);(1,3)(2,5,6,8)(4,7)(9,11)(10,15,18,23)(12,20)(13,21)(14,22)(16,19,17,24);(10,16)(12,13)(15,19)(17,18)(20,21)(23,24);(2,6)(3,7)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=ghi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72235984, #72310728, #72310746, … |
| maximal normal subgroup of | #72235984, #72310728, #72310746, … |
| maximal quotient of | #72311018, #72316715, #72316735 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105656);