Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1723 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)(7,10)(8,13)(9,12)(11,16)(14,17)(15,20)(18,19)(21,22,26,24)(23,29,30,27)(25,32,33,28)(31,34,36,35)(37,38,42,45)(39,43,40,47)(41,44,49,52)(46,50,48,51);(6,7)(9,17)(11,13)(15,18)(23,30)(25,33)(27,29)(28,32)(37,39,42,40)(38,43,45,47)(41,48,49,46)(44,51,52,50);(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,20,18,19)(21,23)(22,27)(24,29)(25,31)(26,30)(28,34)(32,35)(33,36)(37,40,42,39)(38,43,45,47)(41,48,49,46)(44,50,52,51);(1,2)(3,4)(5,8,10,16)(6,11,7,13)(9,18,17,15)(12,20,14,19)(21,24,26,22)(23,27,30,29)(25,32,33,28)(31,34,36,35)(37,38)(39,43)(40,47)(41,44)(42,45)(46,50)(48,51)(49,52);(5,9)(6,12)(7,14)(8,15)(10,17)(11,19)(13,20)(16,18)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,52)(47,51)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fgh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5527496);