Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17093 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,17,7,10,23,4)(3,11,14,8,15,12,28,25)(5,18,20,27,21,9,22,16)(13,26,19,30,31,29,32,24);(1,3,6,14,7,15,23,28)(2,8,4,11,10,25,17,12)(5,19,20,31,21,32,22,13)(9,26,27,24,18,29,16,30);(1,4,6,10,7,17,23,2)(3,12,14,8,15,11,28,25)(5,9,20,27,21,18,22,16)(13,30,19,26,31,24,32,29);(1,5)(2,9)(3,13)(4,16)(6,22)(7,21)(8,24)(10,18)(11,26)(12,29)(14,32)(15,31)(17,27)(19,28)(20,23)(25,30)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,g^d=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72251686, #72251693, #72251708, … |
| maximal normal subgroup of | #72251686, #72251693, #72251708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383803);