Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,24)(20,23);(9,11)(10,14)(12,19,20,17)(13,18)(15,21,23,24)(16,22);(2,4)(9,12)(10,15)(11,17)(13,20)(14,21)(16,23)(18,19)(22,24);(11,18)(14,22)(17,19)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=e,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=di,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265298, #72265370, #72265391, … |
| maximal normal subgroup of | #72265298, #72265370, #72265391, … |
| maximal quotient of | #72266007, #72266032 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264957);