Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,26)(21,25)(23,28)(24,29)(27,31)(30,32);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(19,25)(22,28)(26,31)(29,32);(1,4,5,3)(2,8,6,7)(19,25)(22,28)(26,31)(29,32);(18,23)(22,28)(24,30)(29,32);(17,20,21,27)(18,24,23,30)(19,26,25,31)(22,29,28,32)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=g,f^2=i,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72293016, #72293019, #72293021, … |
| maximal normal subgroup of | #72293016, #72293019, #72293021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535859);