Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | D4×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,11,14,16,21,18)(12,15,20,17,19,22,24,23);(1,2,4,6)(3,8,7,5);(9,11,13,16,14,18,21,10)(12,17,20,22,19,23,24,15);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24);(12,19)(15,22)(17,23)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=ci,f^a=fh,f^b=fh,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72287778, #72319997, #72320183 |
| maximal normal subgroup of | #72287778, #72319997, #72320183 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253261);