Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,17,19,14)(12,15)(13,16)(18,21,24,23)(20,22);(5,6,8,7)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(2,4)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24);(5,7,8,6)(10,16)(14,17)(15,22)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,b^a=be,c^a=cf,d^a=di,e^a=ei,f^b=fi,g^b=gi,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72240659, #72240665, #72240674, … |
| maximal normal subgroup of | #72240659, #72240665, #72240674, … |
| maximal quotient of | #72249539 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261980);