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 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,6)(3,4,7,8)(10,14)(12,20)(15,18)(19,23)(25,26,30,27)(28,33,35,31)(29,34,37,32)(36,38,40,39)(41,42,45,46)(43,47,49,50)(44,48,52,53)(51,54,55,56);(1,3,5,7)(2,4,6,8)(10,14)(11,17)(16,21)(19,23)(28,35)(31,33)(36,40)(38,39)(42,46)(43,49)(44,52)(54,56);(1,4,5,8)(2,3,6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38)(41,43)(42,47)(44,51)(45,49)(46,50)(48,54)(52,55)(53,56);(1,3)(2,4)(5,7)(6,8)(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)(41,42)(43,50)(44,48)(45,46)(47,49)(51,56)(52,53)(54,55);(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)(41,44)(42,48)(43,51)(45,52)(46,53)(47,54)(49,55)(50,56)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgh,c^2=ghi,d^2=hi,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, 6038434);