Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #248455 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #36102 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7)(3,5,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,24)(20,22)(21,23);(2,5)(3,8)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,24)(17,22)(18,23);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,23,24);(2,5)(3,8)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,22,23,24)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,e^d=eh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72249888, #72257416, #72257511, … |
| maximal normal subgroup of | #72249888, #72257416, #72257511, … |
| maximal quotient of | #72257301, #72263993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 155688);