Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498941 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2021 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,10)(5,19)(6,9)(7,12)(8,15)(11,22)(14,30)(16,29)(17,28)(18,26)(20,24)(21,27)(23,31)(25,32);(1,3)(2,8)(4,14)(5,20)(6,16)(7,17)(9,25)(10,23)(11,28)(12,24)(13,19)(15,22)(18,32)(21,31)(26,30)(27,29);(2,9)(4,6)(8,23)(10,12)(11,26)(13,29)(14,16)(15,20)(18,21)(19,27)(24,25)(28,30);(1,4,7,6)(2,10,12,9)(3,14,17,16)(5,18,22,21)(8,23,24,25)(11,27,19,26)(13,30,28,29)(15,31,20,32)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,e^b=ei,h^b=hi,d^c=di,e^c=eh,f^c=fi,g^e=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72213538, #72231746, #72236759, … |
| maximal normal subgroup of | #72213538, #72231746, #72236890, … |
| maximal quotient of | C2×#498943, #72280657, #72280792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406139);