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 | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,7)(10,14)(11,17)(16,21)(19,23)(25,26,30,27)(28,33,35,31)(29,34,37,32)(36,38,40,39)(41,42)(43,44)(45,53)(46,47)(48,54)(49,52)(50,56)(51,55);(1,2,4,5)(3,7,8,6)(10,14)(15,18)(16,21)(22,24)(28,35)(31,33)(36,40)(38,39)(42,47)(43,49)(45,54)(51,56);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38)(41,43,46,49)(42,44,47,52)(45,50,54,55)(48,51,53,56);(1,3,4,8)(2,6,5,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39)(41,44)(42,43)(45,51)(46,52)(47,49)(48,50)(53,55)(54,56);(1,2)(3,6)(4,5)(7,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)(41,45)(42,48)(43,50)(44,51)(46,54)(47,53)(49,55)(52,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=fgh,c^2=ghi,d^2=fghi,e^2=fh,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, 6038554);