Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,8,7)(9,10,12,14)(11,15,16,13)(18,21)(19,23)(25,26,28,30)(27,31,32,29);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,18,20,21)(19,22,23,24)(25,27)(26,29)(28,32)(30,31);(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,24)(20,21)(22,23)(25,26)(27,31)(28,30)(29,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=fh,b^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72260787, #72260818, #72260830, … |
| maximal normal subgroup of | #72260787, #72260818, #72260830, … |
| maximal quotient of | #72328544, #72328558, #72330398 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2524444);