Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667652 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36096 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2047 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)(3,7)(4,6)(5,8)(17,18)(19,26)(20,25)(21,23)(22,29)(24,28)(27,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,27)(25,29)(28,30);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(17,19,21,27)(18,24,23,31)(20,28,22,30)(25,32,29,26);(2,6)(5,7)(18,23)(24,31)(25,29)(26,32);(10,14)(13,15)(17,20,21,22)(18,25,23,29)(19,28,27,30)(24,32,31,26)Polycyclic presentation:
2^9|c^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72283085, #72283138, #72283620 |
| maximal normal subgroup of | #72283085, #72283138, #72283620 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6660227);