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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1713 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)(5,6,10,9)(7,11,13,15)(8,12,16,17)(14,18,19,20)(21,22)(23,32)(24,29)(25,30)(26,40)(27,31)(28,35)(33,45)(34,42)(36,46)(37,43)(38,44)(39,50)(41,49)(47,52)(48,51);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,23,25,34,27,36,26,35)(22,28,40,46,31,42,30,32)(24,33,38,48,37,47,39,49)(29,41,50,52,43,51,44,45);(2,4)(7,13)(11,15)(14,19)(18,20)(21,24)(22,29)(23,33)(25,38)(26,39)(27,37)(28,41)(30,44)(31,43)(32,45)(34,48)(35,49)(36,47)(40,50)(42,51)(46,52);(5,8,10,16)(6,12,9,17)(7,14,13,19)(11,18,15,20)(24,37)(29,43)(33,47)(38,39)(41,51)(44,50)(45,52)(48,49);(5,9,10,6)(7,15,13,11)(8,12,16,17)(14,18,19,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,38,37,39)(28,32,42,46)(29,44,43,50)(33,48,47,49)(41,45,51,52)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473020);