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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1724 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)(26,41)(27,31)(28,35)(33,48)(34,43)(36,49)(37,52)(38,45)(39,46)(40,47)(42,50)(44,51);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,23,25,34,27,36,26,35)(22,28,41,49,31,43,30,32)(24,37,39,50,40,33,38,51)(29,44,45,48,47,42,46,52);(2,4)(7,13)(11,15)(14,19)(18,20)(21,24)(22,29)(23,33)(25,39)(26,38)(27,40)(28,42)(30,46)(31,47)(32,48)(34,51)(35,50)(36,37)(41,45)(43,44)(49,52);(5,8,10,16)(6,12,9,17)(7,14,13,19)(11,18,15,20)(21,25,27,26)(22,30,31,41)(23,34,36,35)(24,38,40,39)(28,32,43,49)(29,45,47,46)(33,50,37,51)(42,52,44,48);(5,9,10,6)(7,15,13,11)(8,12,16,17)(14,18,19,20)(21,25,27,26)(22,30,31,41)(23,34,36,35)(24,39,40,38)(28,32,43,49)(29,46,47,45)(33,51,37,50)(42,48,44,52)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473030);