Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457092 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1418 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)(9,10,14,17)(11,18,16,13)(12,15,23,20)(19,22,24,21);(5,6,7,8)(9,11,13,10,14,16,18,17)(12,20,22,24,23,15,21,19);(2,4)(6,8)(9,12)(10,15)(11,19)(13,21)(14,23)(16,24)(17,20)(18,22);(12,21,23,22)(15,24,20,19)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214416, #72215354, #72216148, … |
| maximal normal subgroup of | #72214416, #72215354, #72216148, … |
| maximal quotient of | #72214149, #72215281 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382961);