Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34643 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,25,30,26,29,37,44)(23,27,32,40,33,39,47,51)(24,28,36,43,35,42,45,49)(31,38,46,50,34,41,48,52);(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,14)(18,20)(21,23,25,32,26,33,37,47)(22,27,30,40,29,39,44,51)(24,34,36,48,35,31,45,46)(28,41,43,52,42,38,49,50);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24)(22,28)(23,31)(25,36)(26,35)(27,38)(29,42)(30,43)(32,46)(33,34)(37,45)(39,41)(40,50)(44,49)(47,48)(51,52);(7,16)(11,14)(15,17)(19,20)(22,29)(24,35)(27,39)(30,44)(31,34)(36,45)(40,51)(46,48);(5,9,10,8)(6,12,13,18)(7,17,16,15)(11,19,14,20)(24,35)(28,42)(31,34)(36,45)(38,41)(43,49)(46,48)(50,52)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6296809);