Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6374204 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2308 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12,14,7)(4,16,18,9)(5,11)(6,10)(8,17,26,20)(13,23)(15,28,19,24)(21,27)(22,31)(25,32)(29,30);(1,3)(2,7)(4,13)(5,19)(6,14)(8,25)(9,23)(10,12)(11,28)(15,21)(16,29)(17,32)(18,30)(20,31)(22,26)(24,27);(1,4)(2,8)(3,13)(5,20)(6,18)(7,22)(9,11)(10,26)(12,25)(14,30)(15,32)(16,27)(17,21)(19,31)(23,24)(28,29);(1,4)(2,9)(3,13)(5,17)(6,18)(7,23)(8,11)(10,16)(12,29)(14,30)(15,31)(19,32)(20,21)(22,24)(25,28)(26,27);(3,14)(7,12)(13,30)(15,19)(22,25)(23,29)(24,28)(31,32)Polycyclic presentation:
2^9|a^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,c^b=cf,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72233776, #72233777, #72233779, … |
| maximal normal subgroup of | #72233776, #72233777, #72233779, … |
| maximal quotient of | #72317896, #72318156, #72318255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281400);