Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3106 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)(3,5)(4,7)(6,8)(10,13)(11,15)(18,20)(19,23)(27,28)(31,32);(2,3)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,23,24)(25,26,30,29)(27,32,28,31);(2,3)(6,7)(10,13)(14,16)(19,23)(22,24)(25,27,30,28)(26,31,29,32);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)(25,28,30,27)(26,32,29,31);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,24,23,22)(25,29,30,26)(27,32,28,31)Polycyclic presentation:
2^9|b^2=ghi,c^2=fi,d^2=g,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72336309, #72336381, #72336445, … |
| maximal normal subgroup of | #72336309, #72336381, #72336445, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 894709);