Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,24,22,25,20,29)(19,23,28,31,26,30,27,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32);(19,26)(23,30)(27,28)(31,32);(17,20,22,21)(18,24,25,29)(19,27,26,28)(23,31,30,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,g^2=i,b^a=bf,d^a=di,f^a=fh,c^b=ci,g^b=gh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239607, #72239635, #72239721, … |
| maximal normal subgroup of | #72239607, #72239635, #72239721, … |
| maximal quotient of | #72239619, #72239873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6249835);