Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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:
(4,8)(6,7)(10,14)(12,19)(15,20)(18,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40)(41,42,45,46)(43,49,51,47)(44,52,54,48)(50,55,53,56);(1,2,5,3)(4,6,8,7)(10,14)(11,17)(16,21)(18,23)(26,27)(28,35)(32,34)(36,40)(42,46)(47,49)(48,52)(55,56);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39)(41,43)(42,47)(44,53)(45,51)(46,49)(48,56)(50,54)(52,55);(1,4,5,8)(2,6,3,7)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40)(41,42)(43,47)(44,48)(45,46)(49,51)(50,55)(52,54)(53,56);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)(41,44)(42,48)(43,50)(45,54)(46,52)(47,55)(49,56)(51,53)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=fi,d^2=fhi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4637016);