Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200400 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #34977 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,14,18,15,17,13,11)(12,20,21,24,22,16,23,19);(1,2,3,5,4,6,7,8)(9,11,14,17,15,18,13,10)(12,19,23,20,22,24,21,16);(9,12)(10,16)(11,19)(13,21)(14,23)(15,22)(17,20)(18,24);(2,6)(5,8)(9,13,15,14)(10,11,17,18)(12,21,22,23)(16,19,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282326, #72282373, #72283314 |
| maximal normal subgroup of | #72282326, #72282373, #72283314 |
| maximal quotient of | #72282907 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107724);