Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | |
1797 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)(26,40)(27,31)(28,34)(33,47)(35,42)(36,44)(37,45)(38,43)(39,46)(41,49)(48,51)(50,52);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,38,27,35,26,34)(22,28,40,42,31,43,30,32)(24,33,39,48,37,50,36,49)(29,41,44,52,45,51,46,47);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,39)(26,36)(27,37)(28,41)(30,46)(31,45)(32,47)(34,49)(35,50)(38,48)(40,44)(42,52)(43,51);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,31,40)(23,34,35,38)(24,36,37,39)(28,42,43,32)(29,44,45,46)(33,48,50,49)(41,47,51,52);(8,15)(11,18)(14,19)(17,20)(23,35)(24,37)(28,43)(29,45)(32,42)(34,38)(36,39)(44,46)Polycyclic presentation:
2^9|a^2=h,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=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6471673);