Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60750 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,21,25,22,24,30,31)(19,23,27,20,26,28,32,29);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19)(18,23)(20,25)(21,27)(22,26)(24,28)(29,31)(30,32);(17,20,21,28,22,29,30,23)(18,19,25,27,24,26,31,32);(18,24)(19,26)(25,31)(27,32);(2,7)(6,15)(8,12)(9,16)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,f^2=h,g^2=i,b^a=bf,c^a=ci,d^a=di,e^a=eh,f^a=fh,d^b=di,f^b=fh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72306663, #72306669, #72306683, … |
| maximal normal subgroup of | #72306663, #72306669, #72306683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252060);