Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1867 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)(3,7)(4,6)(5,8)(17,18)(19,21)(20,22)(23,24);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14);(3,8)(5,7)(10,13)(14,15)(18,21)(19,24)(22,23);(2,6)(5,7);(9,11,12,16)(10,14,13,15)(18,22)(21,23)Polycyclic presentation:
2^9|b^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282016, #72305302, #72305310 |
| maximal normal subgroup of | #72282016, #72305302, #72305310 |
| maximal quotient of | #72305740, #72305757, #72316322 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6707452);