Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10564 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,5)(4,6)(7,8)(10,13,14,15);(1,2)(3,7,8,5)(4,6);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,b^a=bd,c^a=ce,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=ei,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72233810, #72233815, #72233816, … |
| maximal normal subgroup of | #72233810, #72233815, #72233816, … |
| maximal quotient of | #72279561 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23609);