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 | |
1463 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)(27,38)(28,32)(29,34)(30,35)(31,37)(33,41)(36,39)(40,53)(42,47)(43,48)(44,50)(45,52)(46,51)(49,55)(54,56);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,39,43,31,41,30,28)(26,32,35,33,37,48,36,38)(29,40,44,54,46,55,45,42)(34,47,52,49,51,56,50,53);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,41,31,43,39,27)(26,33,36,32,37,38,35,48)(29,42,45,55,46,54,44,40)(34,49,50,47,51,53,52,56);(12,20)(15,24)(17,19)(21,23)(25,29,31,46)(26,34,37,51)(27,40,41,55)(28,42,43,54)(30,45,39,44)(32,47,48,56)(33,49,38,53)(35,52,36,50);(11,18)(14,22)(17,19)(21,23)(25,30,31,39)(26,35,37,36)(27,28,41,43)(29,44,46,45)(32,33,48,38)(34,50,51,52)(40,54,55,42)(47,53,56,49)Polycyclic presentation:
2^9|a^2=h,b^2=gi,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=dh,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6417968);