Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3038 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(5,6,8,7)(9,10)(11,14)(12,20)(13,17)(15,19)(16,21)(18,23)(22,24);(1,3,4,2)(6,7)(9,11,14,10)(12,18,22,16)(13,17,15,19)(20,21,24,23);(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20);(5,7,8,6)(10,11)(16,18)(17,19)(20,24);(9,13,14,15)(10,17,11,19)(12,21,22,23)(16,24,18,20)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,d^b=df,e^b=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72283123, #72284234, #72284863, … |
| maximal normal subgroup of | #72283123, #72284234, #72284863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289062);