Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #411390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15115 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
756 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)(4,17)(5,6)(10,24)(13,26)(14,28)(15,20)(18,19)(22,25)(23,27)(29,30)(31,32);(1,2,6,12,7,8,5,11)(3,13,15,21,16,26,20,9)(4,10,17,25,18,22,19,24)(14,27,28,31,29,32,30,23);(1,3)(2,9,8,21)(4,14)(5,15)(6,20)(7,16)(10,23,22,31)(11,13,12,26)(17,30)(18,29)(19,28)(24,27,25,32);(1,4,5,19,7,18,6,17)(2,10,11,24,8,22,12,25)(3,14,15,28,16,29,20,30)(9,23,13,27,21,31,26,32)Polycyclic presentation:
2^9|b^2=h,c^2=e,d^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251684, #72251692, #72251699, … |
| maximal normal subgroup of | #72251684, #72251692, #72251699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318586);