Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1587 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)(7,13)(8,15)(11,16)(12,17)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33)(37,38,43,47)(39,45,40,44)(41,52,46,42)(48,49,50,51);(6,10)(11,16)(12,17)(18,20)(24,32)(27,34)(30,36)(33,35)(37,39,43,40)(38,44,47,45)(41,48,46,50)(42,49,52,51);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36)(37,40,43,39)(38,45,47,44)(41,48,46,50)(42,49,52,51);(1,2,3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35)(37,41,43,46)(38,42,47,52)(39,48,40,50)(44,49,45,51);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)(37,42)(38,46)(39,49)(40,51)(41,47)(43,52)(44,50)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=hi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5417698);