Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,4,7,6)(11,15)(14,16)(17,18)(19,20)(21,23)(22,24)(25,26,28,30)(27,31,32,29);(1,3)(2,6)(4,8)(5,7)(9,10,13,12)(11,14,15,16)(17,19)(18,22)(20,23)(21,24);(17,18)(19,20)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,24)(21,23)(25,27,28,32)(26,29,30,31);(1,4)(2,7)(3,8)(5,6)(9,12,13,10)(11,16,15,14)(17,20,21,22)(18,19,23,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,d^2=g,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72304048, #72304093, #72326808, … |
| maximal normal subgroup of | #72304048, #72304093, #72326808, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1269358);