Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1238 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)(6,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26)(27,28)(29,34)(30,31)(32,33)(35,36)(37,40)(38,39)(41,42)(43,48)(44,50)(45,47)(46,49)(51,55)(52,56)(53,54);(11,18)(12,21)(14,17)(15,20)(26,31)(27,33)(34,39)(35,37)(41,43,47,44)(42,48,45,50)(46,52,54,51)(49,56,53,55);(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27)(26,32)(28,31)(29,37)(30,33)(34,36)(35,38)(39,40)(41,44,47,43)(42,48,45,50)(46,52,54,51)(49,55,53,56);(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,21)(22,24)(25,28,30,32)(26,33,31,27)(29,36,38,40)(34,37,39,35)(41,45,47,42)(43,50,44,48)(46,53,54,49)(51,56,52,55);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,19)(13,21)(14,22)(16,20)(17,23)(18,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,40)(33,37)(41,46,47,54)(42,49,45,53)(43,51,44,52)(48,55,50,56)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=gh,d^2=fgh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5479596);