Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6346762 |
| 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 |
| prime radical | #62818 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,8)(4,9)(5,7)(6,10)(11,15)(12,16)(13,14)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28);(2,7)(4,5)(8,14)(9,10)(12,13)(15,16)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,16)(10,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(3,11)(8,15)(12,13)(14,16);(1,4,6,5)(2,7,10,9)(3,12,11,13)(8,14,15,16)Polycyclic presentation:
2^9|a^2=i,e^2=h,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307976, C2×#6346774, #72310771, … |
| maximal normal subgroup of | #72307976, C2×#6346774, #72310771, … |
| maximal quotient of | #72308083, C2×#6346774, #72309223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253970);