Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #112984 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#3109 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,8,4,14)(3,10,11,16,12,7,13,15)(17,18,21,25)(19,26)(20,22,30,24)(23,29)(27,31)(28,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,10)(9,16)(14,15)(18,22)(19,27)(20,30)(24,25)(28,29)(31,32);(2,8)(3,11)(4,5)(7,15)(10,16)(12,13)(17,19)(18,23)(20,28)(21,29)(22,31)(24,32)(25,26)(27,30)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,f^c=fh,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72238999, #72249689, #72249691, … |
| maximal normal subgroup of | #72238999, #72249689, #72249691, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20185);