Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34935 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2194 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)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,30)(26,31)(27,29)(28,32);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,20,26)(18,22,23,31)(21,25,28,27)(24,30,32,29);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(18,23)(19,25)(21,28)(22,29)(26,27)(30,31);(1,4,5,3)(2,8,6,7)(19,25)(21,28)(22,30)(24,32)(26,27)(29,31);(19,26)(22,31)(25,27)(29,30)Polycyclic presentation:
2^9|b^2=g,c^2=fgi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281059, #72281103, #72281204, … |
| maximal normal subgroup of | #72281059, #72281103, #72281204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354216);