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 | |
1232 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)(3,5)(4,8)(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)(42,46)(43,49)(48,51)(50,55);(2,7)(4,5)(10,14)(15,18)(16,21)(22,24)(28,35)(31,33)(36,40)(38,39)(41,42,45,46)(43,47,49,53)(44,51,52,48)(50,56,55,54);(1,3)(2,4)(5,7)(6,8)(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)(41,42)(43,47)(44,48)(45,46)(49,53)(50,54)(51,52)(55,56);(1,4,6,5)(2,8,7,3)(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)(41,43,45,49)(42,47,46,53)(44,50,52,55)(48,54,51,56);(1,5,6,4)(2,3,7,8)(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)(41,44)(42,48)(43,50)(45,52)(46,51)(47,54)(49,55)(53,56)Polycyclic presentation:
2^9|a^2=hi,b^2=fgh,c^2=fghi,d^2=ghi,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, 6038549);