Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| prime radical | #62686 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10,13,21,14,17,12,16)(11,18,19,24,20,15,22,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,21)(14,17)(18,20)(19,23)(22,24);(2,5)(6,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24);(1,3,4,7)(2,6,5,8);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)Polycyclic presentation:
2^9|a^2=fg,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342604);