Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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:
(3,7)(5,8)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,37)(38,40)(41,42)(43,49)(44,52)(45,48)(46,51)(47,53)(50,54)(55,56);(11,18)(14,17)(19,24)(21,23)(26,28)(29,37)(31,33)(34,40)(41,42,45,48)(43,46,51,49)(44,52,53,47)(50,55,56,54);(1,2,4,6)(3,5,7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39)(41,42)(43,49)(44,47)(45,48)(46,51)(50,55)(52,53)(54,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,18)(10,14,16,17)(12,19,20,24)(15,21,22,23)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39)(41,43)(42,46)(44,50)(45,51)(47,54)(48,49)(52,55)(53,56);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)(41,44,45,53)(42,47,48,52)(43,50,51,56)(46,54,49,55)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fgi,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4401921);