Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4546 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)(9,10,13,16,14,18,12,17)(11,19,21,23,22,24,20,15);(5,6)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,24)(18,23);(2,4)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,20)(21,24)(22,23);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,23,24)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,c^b=cg,e^b=ei,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219669, #72239564, #72239576, … |
| maximal normal subgroup of | #72219669, #72239564, #72239576, … |
| maximal quotient of | #72219664, #72240256, #72249789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264686);