Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382865 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14345 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,19,7)(3,12,14,20,15,8,22,24)(4,9,17,26,18,27,32,21)(13,28,16,30,29,23,31,25);(2,7)(5,19)(8,20)(9,21)(10,11)(12,24)(14,22)(16,31)(17,32)(23,30)(25,28)(26,27);(1,3)(2,8,11,12)(4,16,18,31)(5,14)(6,15)(7,20,10,24)(9,25)(13,17,29,32)(19,22)(21,23)(26,28)(27,30);(1,4)(2,9)(3,13,15,29)(5,17)(6,18)(7,21)(8,23,12,28)(10,26)(11,27)(14,16,22,31)(19,32)(20,30,24,25)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=f,e^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,e^b=ei,d^c=de,f^c=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216037, #72216077, #72216181, … |
| maximal normal subgroup of | #72216037, #72216077, #72216181, … |
| maximal quotient of | C2×#382865, #72252158, #72255607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290061);