Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201928 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2192 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)(3,7,8,5)(4,6)(9,10,13,17,14,18,12,15)(11,19,20,16,22,23,21,24);(1,2)(3,5)(4,6)(7,8)(10,15,18,17)(16,19,24,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,20)(14,22)(15,19)(17,23)(18,24);(9,12,14,13)(10,15,18,17)(11,20,22,21)(16,23,24,19)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72231222, #72236668, #72236696, … |
| maximal normal subgroup of | #72231222, #72236668, #72236696, … |
| maximal quotient of | #72231177, #72231376, #72263333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109203);