Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #301917 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3505 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
973 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,7)(5,14)(6,10)(8,12)(9,15)(13,16)(17,18,21,24)(19,22,26,30)(20,23,28,31)(25,32,27,29);(1,3,5,4,6,12,15,13)(2,7,9,11,10,16,14,8)(17,19)(18,22)(20,27)(21,26)(23,29)(24,30)(25,28)(31,32);(17,20)(18,23)(19,25,26,27)(21,28)(22,29,30,32)(24,31);(1,4,5,12,6,13,15,3)(2,8,9,7,10,11,14,16)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^c=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72268348, #72268351, #72268367, … |
| maximal normal subgroup of | #72268348, #72268351, #72268367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 209121);