Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #301861 |
| 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 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,13)(5,15)(6,10)(7,11)(9,14)(12,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,30)(28,32)(29,31);(1,3,5,11,6,12,14,4)(2,7,9,8,10,13,15,16)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,30,31,32);(17,20)(18,23)(19,26,27,28)(21,29)(22,30,25,32)(24,31);(1,4,14,12,6,11,5,3)(2,8,15,7,10,16,9,13)(18,24)(22,25)(23,31)(30,32)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=ehi,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72268336, #72268351, #72268375, … |
| maximal normal subgroup of | #72268336, #72268351, #72268375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 209081);