Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1641 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)(21,22,25,26)(23,29,31,27)(24,28,32,33)(30,35,36,34)(37,38)(39,45)(40,48)(41,44)(42,47)(43,49)(46,50)(51,52);(6,10)(11,16)(12,17)(18,20)(22,26)(24,32)(27,29)(30,36)(37,38,41,44)(39,42,47,45)(40,48,49,43)(46,51,52,50);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,49)(50,52);(1,2,3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36)(37,39)(38,42)(40,46)(41,47)(43,50)(44,45)(48,51)(49,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36)(37,40)(38,43)(39,46)(41,49)(42,50)(44,48)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4410254);