Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499844 |
| 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 | #60558 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(18,23)(19,27)(20,21)(25,26)(29,30)(31,32);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,26)(19,28,30,24)(20,25,21,23)(27,31,29,32);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,30)(18,24,26,28)(20,27,21,29)(23,31,25,32);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20,22,21)(18,25,26,23)(19,29,30,27)(24,31,28,32);(2,8)(6,9)(7,16)(14,15)(18,26)(23,25)(24,28)(31,32)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311102 |
| maximal normal subgroup of | #72311102 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436238);