Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417647 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3003 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2512 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)(9,10,12,16,13,15,21,23)(11,17,19,22,20,24,18,14);(5,6)(7,8)(9,11)(10,14)(12,18)(13,20)(15,22)(16,24)(17,23)(19,21);(2,4)(6,8);(10,15)(11,18)(12,21)(14,17)(19,20)(22,24)Polycyclic presentation:
2^9|a^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321910, #72321993 |
| maximal normal subgroup of | #72321910, #72321993 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326015);