Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #53065 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1459 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)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26)(27,31)(28,33)(29,32)(30,35)(34,42)(36,44)(37,43)(38,46)(39,48)(40,49)(41,47)(45,52)(50,54)(51,55)(53,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,30,38)(26,31,35,46)(28,39,40,36)(29,37,42,52)(32,43,34,45)(33,48,49,44)(41,53,54,51)(47,56,50,55);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(29,42)(31,43)(33,47)(34,35)(37,52)(41,54)(44,55)(45,46)(48,56)(49,50)(51,53);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,28,30,40)(26,33,35,49)(27,36,38,39)(29,41,42,54)(31,44,46,48)(32,47,34,50)(37,51,52,53)(43,55,45,56);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(28,40)(33,49)(36,39)(41,54)(44,48)(47,50)(51,53)(55,56)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7108532);