Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #239395 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60958 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
980 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(18,23)(19,27)(20,21)(24,32)(25,26)(29,30);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,18)(19,28)(20,25)(21,23)(22,26)(24,29)(27,32)(30,31);(4,11)(6,12)(7,14)(9,15)(17,19,20,27,22,30,21,29)(18,24,23,31,26,32,25,28);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,20,22,21)(18,25,26,23)(19,27,30,29)(24,28,32,31)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,d^c=dhi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72315941, #72315946, #72316018, … |
| maximal normal subgroup of | #72315941, #72315946, #72316018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146593);