Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34977 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,23,28,24,29,22,32)(19,26,21,27,20,30,31,25);(2,6)(7,8)(17,19,24,20)(18,25,29,27)(21,23,31,22)(26,28,30,32);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,31,24,19,22,21)(18,26,28,27,29,30,32,25);(1,4,5,3)(2,8,6,7)(17,21,23,20,24,31,22,19)(18,27,28,30,29,25,32,26);(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,28,29,32)(19,21,20,31)(25,30,27,26)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282648, #72282738, #72282893, … |
| maximal normal subgroup of | #72282648, #72282738, #72282893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366425);