Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2358 |
| prime radical | #18516 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
802 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,5)(4,6,8,7)(9,10,14,18,15,19,13,24)(11,12,20,16,17,23,21,22);(2,5)(6,7)(9,11,13,21,15,17,14,20)(10,16,24,12,19,22,18,23);(2,6)(4,8)(5,7)(9,12)(10,17)(11,19)(13,22)(14,16)(15,23)(18,21)(20,24);(9,11)(10,12)(13,21)(14,20)(15,17)(16,18)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dehi,e^b=eh,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72286208, #72286234, #72286392 |
| maximal normal subgroup of | #72286208, #72286234, #72286392 |
| maximal quotient of | #72287279, #72287378, #72287422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 135446);