Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17577 |
| automorphism | ? |
| 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 | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3286 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)(3,10,13,7)(4,8)(5,14)(6,9)(11,16,12,15)(17,18)(19,27,29,32)(20,25)(21,26)(22,24)(23,28,31,30);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(18,24)(23,31)(25,26)(27,32);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16);(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,27,31,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=ci,d^a=dh,f^a=fh,f^b=fi,g^b=ghi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72284121, #72306325, #72306329, … |
| maximal normal subgroup of | #72284121, #72306325, #72306329, … |
| maximal quotient of | #72308146, C2×#6343376, #72309244, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250572);