Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17239 |
| 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 | |
647 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11)(3,13)(4,18,21,26)(5,10,6,12)(8,16)(9,20,25,19)(14,28)(15,24)(17,23)(22,31)(27,29)(30,32);(1,3,7,17)(2,8)(4,14)(5,15,6,16)(9,22,25,28)(10,13)(11,24)(12,23)(18,27,26,32)(19,29)(20,30)(21,31);(1,4,6,19,7,21,5,20)(2,9,10,18,11,25,12,26)(3,14,16,29,17,31,15,30)(8,22,13,27,24,28,23,32);(2,10)(4,19)(5,6)(8,23)(11,12)(13,24)(14,29)(15,16)(18,26)(20,21)(22,28)(30,31)Polycyclic presentation:
2^9|a^2=g,b^2=eh,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,h^c=hi,e^d=ei,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251686, #72251692, #72251706, … |
| maximal normal subgroup of | #72251686, #72251692, #72251706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386964);