Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347145 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11)(4,15)(5,9)(6,8)(7,13)(10,18)(12,25)(14,26)(16,21)(17,30)(19,23)(20,31)(22,29)(24,32)(27,28);(1,3)(2,7)(4,16)(5,12)(6,14)(8,22)(9,21)(10,25)(11,15)(13,18)(17,31)(19,28)(20,23)(24,27)(26,30)(29,32);(4,17)(10,24)(11,26)(13,29)(15,30)(16,31)(18,32)(25,27);(2,8)(3,12)(7,20)(9,23)(10,24)(11,27)(13,16)(14,28)(15,30)(21,22)(25,26)(29,31);(2,9)(7,21)(8,23)(10,15)(11,25)(20,22)(24,30)(26,27)Polycyclic presentation:
2^9|f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,d^b=dh,f^b=fh,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283025, #72283074, #72283632, … |
| maximal normal subgroup of | #72283025, #72283074, #72283632, … |
| maximal quotient of | #72284499 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254341);