Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,32)(24,29)(25,30)(26,41)(27,31)(28,34)(33,49)(35,43)(36,52)(37,46)(38,47)(39,44)(40,48)(42,51)(45,50);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,39,27,35,26,34)(22,28,41,43,31,44,30,32)(24,36,40,51,38,33,37,50)(29,45,46,49,47,42,48,52);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,40)(26,37)(27,38)(28,42)(30,48)(31,47)(32,49)(34,51)(35,36)(39,50)(41,46)(43,52)(44,45);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,31,41)(23,34,35,39)(24,37,38,40)(28,43,44,32)(29,46,47,48)(33,50,36,51)(42,49,45,52);(8,15)(11,18)(14,19)(17,20)(23,35)(24,38)(28,44)(29,47)(32,43)(34,39)(37,40)(46,48)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^c=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6471682);