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 | |
1236 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)(3,7)(9,10,14,11)(12,17,19,15)(13,18,21,16)(20,22,24,23)(25,26,30,31)(27,34,32,28)(29,33,37,38)(35,40,39,36)(42,43)(47,49)(48,50)(54,55);(2,5)(6,8)(12,19)(15,17)(20,24)(22,23)(26,31)(27,32)(29,37)(36,40)(41,42,46,43)(44,49,52,47)(45,50,53,48)(51,54,56,55);(1,2,4,5)(3,8,7,6)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,27,30,32)(26,28,31,34)(29,35,37,39)(33,36,38,40)(41,43,46,42)(44,49,52,47)(45,50,53,48)(51,55,56,54);(1,3,4,7)(2,6,5,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,28)(26,32)(27,31)(29,36)(30,34)(33,39)(35,38)(37,40)(41,44)(42,47)(43,49)(45,51)(46,52)(48,54)(50,55)(53,56);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)(41,45,46,53)(42,48,43,50)(44,51,52,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=ghi,d^2=fghi,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, 6038312);