Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405381 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14912 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(5,6)(9,13)(10,21)(11,20)(17,19)(22,31)(23,25)(24,30)(27,32)(28,29);(1,2)(3,13,12,25)(4,10)(5,20)(6,8)(7,11)(9,16,23,15)(14,27,26,31)(17,30)(18,24)(19,21)(22,29,32,28);(1,3,5,15,7,12,6,16)(2,9,8,13,11,23,20,25)(4,14,17,28,18,26,19,29)(10,22,21,27,24,32,30,31);(1,4)(2,10)(3,14)(5,17)(6,19)(7,18)(8,21)(9,22)(11,24)(12,26)(13,27)(15,28)(16,29)(20,30)(23,32)(25,31)Polycyclic presentation:
2^9|b^2=h,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72319352, #72319375, #72319453 |
| maximal normal subgroup of | #72319352, #72319375, #72319453 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312577);