Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500450 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1467 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)(27,37)(28,32)(29,34)(30,45)(31,36)(33,40)(35,38)(39,51)(41,46)(42,47)(43,49)(44,55)(48,53)(50,52)(54,56);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,38,42,31,40,30,28)(26,32,45,33,36,47,35,37)(29,39,52,54,43,53,44,41)(34,46,55,48,49,56,50,51);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,40,31,42,38,27)(26,33,35,32,36,37,45,47)(29,41,44,53,43,54,52,39)(34,48,50,46,49,51,55,56);(12,20)(15,24)(17,19)(21,23)(25,29)(26,34)(27,39)(28,41)(30,44)(31,43)(32,46)(33,48)(35,50)(36,49)(37,51)(38,52)(40,53)(42,54)(45,55)(47,56);(11,18)(14,22)(17,19)(21,23)(29,43)(34,49)(39,53)(41,54)(44,52)(46,56)(48,51)(50,55)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=f,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=dh,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6417966);