Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498916 |
| 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 | |
2115 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)(3,12)(4,17)(5,8)(6,9)(7,14)(10,21)(11,22)(13,30)(15,29)(16,27)(18,23)(19,28)(20,26)(24,32)(25,31);(1,3)(2,7)(4,18)(5,13)(6,16)(8,25)(9,23)(10,27)(11,24)(12,17)(14,21)(15,22)(19,32)(20,31)(26,29)(28,30);(2,8)(3,13)(5,22)(9,11)(10,26)(12,27)(14,32)(15,16)(17,28)(18,31)(19,20)(24,25);(2,9)(7,23)(8,11)(10,17)(12,27)(24,25)(26,28)(29,30)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=eh,f^c=fi,g^c=gi,g^e=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72236949, #72237532, #72237544, … |
| maximal normal subgroup of | #72236949, #72237532, #72237544, … |
| maximal quotient of | #72312327, C2×#498916, #72313372, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406112);