Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1391 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,29,33,30,34,41,48)(27,31,37,44,35,42,49,53)(28,32,40,47,39,46,52,56)(36,43,50,54,38,45,51,55);(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27)(26,31)(28,38)(29,37)(30,35)(32,45)(33,44)(34,42)(36,39)(40,51)(41,49)(43,46)(47,55)(48,53)(50,52)(54,56);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(27,35)(31,42)(36,38)(37,49)(43,45)(44,53)(50,51)(54,55);(11,20)(15,18)(19,21)(23,24)(25,28)(26,32)(27,36)(29,40)(30,39)(31,43)(33,47)(34,46)(35,38)(37,50)(41,52)(42,45)(44,54)(48,56)(49,51)(53,55);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(27,35)(28,39)(31,42)(32,46)(37,49)(40,52)(44,53)(47,56)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6294679);