Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(4,8)(10,14)(15,18)(16,21)(22,24)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39)(41,42)(43,47)(44,46)(45,48)(49,51)(50,54)(52,53)(55,56);(1,2,5,6)(3,8,7,4)(11,17)(15,18)(19,23)(22,24)(42,44)(43,49)(45,53)(54,56);(1,3,5,7)(2,4,6,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40)(41,43,46,49)(42,47,44,51)(45,50,53,55)(48,54,52,56);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39)(41,44,46,42)(43,47,49,51)(45,52,53,48)(50,54,55,56);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)(41,45,46,53)(42,48,44,52)(43,50,49,55)(47,54,51,56)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=fghi,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6038653);