Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1638 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)(7,11)(8,10)(9,16)(12,18)(13,20)(14,15)(17,19)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34)(37,38,42,45)(39,43,40,47)(41,44,49,52)(46,50,48,51);(6,8)(11,15)(12,16)(19,20)(23,29)(26,33)(30,35)(34,36)(37,39,42,40)(38,43,45,47)(41,48,49,46)(44,51,52,50);(5,7)(6,11)(8,15)(9,17)(10,14)(12,20)(13,18)(16,19)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35)(37,40,42,39)(38,43,45,47)(41,48,49,46)(44,50,52,51);(1,2,3,4)(5,8,10,6)(7,11,14,15)(9,16,18,12)(13,19,17,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36)(37,38)(39,43)(40,47)(41,44)(42,45)(46,50)(48,51)(49,52);(5,9)(6,12)(7,13)(8,16)(10,18)(11,19)(14,17)(15,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,52)(47,51)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=ghi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5191995);