Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18162 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,20)(19,24)(22,23);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,23)(18,21);(3,7)(5,8);(2,6)(5,8)(11,19)(15,23)(18,24)(20,22);(9,12,14,13)(10,16,17,21)(11,20,19,22)(15,24,23,18)Polycyclic presentation:
2^9|e^2=hi,f^2=h,b^a=bf,c^a=cg,f^a=fh,d^b=dh,f^b=fh,g^b=gi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214220, #72214707, #72214821, … |
| maximal normal subgroup of | #72214220, #72214707, #72214821, … |
| maximal quotient of | #72257867 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253651);