Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632108 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #36702 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,9)(4,19)(6,7)(10,25)(11,20)(12,13)(14,27)(15,30)(16,32)(17,24)(21,23)(26,29);(1,2,8,12)(3,14,18,25)(4,5,11,22)(6,13,7,9)(10,24,27,17)(15,31,26,28)(16,29,32,30)(19,23,20,21);(1,3)(2,10)(4,15)(5,16)(6,17)(7,24)(8,18)(9,14)(11,26)(12,27)(13,25)(19,30)(20,29)(21,31)(22,32)(23,28);(1,4,8,11)(2,5,12,22)(3,15,18,26)(6,20,7,19)(9,23,13,21)(10,16,27,32)(14,28,25,31)(17,29,24,30);(1,5)(2,11)(3,16)(4,12)(6,21)(7,23)(8,22)(9,20)(10,26)(13,19)(14,29)(15,27)(17,31)(18,32)(24,28)(25,30)Polycyclic presentation:
2^9|b^2=i,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,f^a=fi,h^a=hi,c^b=ch,d^b=dfh,f^b=fi,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72306460 |
| maximal normal subgroup of | #72306460 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6539308);