Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1442 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)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,32,38)(27,34,41,48)(28,33,40,49)(29,35,46,54)(30,36,31,37)(39,50,55,51)(42,47,43,56)(44,52,45,53);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,40,32,41,30,28)(26,33,36,48,38,49,37,34)(29,39,44,43,46,55,45,42)(35,47,53,51,54,56,52,50);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,41,32,40,31,27)(26,34,37,49,38,48,36,33)(29,43,45,39,46,42,44,55)(35,51,52,47,54,50,53,56);(12,20)(15,24)(17,19)(21,23)(25,29,32,46)(26,35,38,54)(27,39,41,55)(28,42,40,43)(30,45,31,44)(33,47,49,56)(34,50,48,51)(36,53,37,52);(11,18)(14,22)(17,19)(21,23)(25,30,32,31)(26,36,38,37)(27,28,41,40)(29,44,46,45)(33,48,49,34)(35,52,54,53)(39,43,55,42)(47,50,56,51)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6417961);