Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62972 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3544 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 52 points:
(1,2,3,4)(5,6)(7,11)(8,15)(9,10)(12,16)(13,17)(14,20)(18,19)(22,27)(23,31)(29,34)(30,39)(32,43)(33,44)(42,47)(46,51);(6,10)(11,17)(12,15)(18,20)(23,31)(25,37)(28,38)(30,39)(32,43)(35,48)(40,45)(42,47);(7,13)(11,17)(14,19)(18,20)(21,22,26,27)(23,28,31,38)(24,34,36,29)(25,30,37,39)(32,45,43,40)(33,41,44,49)(35,47,48,42)(46,52,51,50);(5,7,9,13)(6,11,10,17)(8,14,16,19)(12,18,15,20)(21,23)(22,28)(24,32)(25,33)(26,31)(27,38)(29,40)(30,41)(34,45)(35,46)(36,43)(37,44)(39,49)(42,50)(47,52)(48,51);(5,6)(7,11)(8,12)(9,10)(13,17)(14,18)(15,16)(19,20)(21,24,26,36)(22,29,27,34)(23,32,31,43)(25,35,37,48)(28,40,38,45)(30,42,39,47)(33,46,44,51)(41,50,49,52);(5,8)(6,12)(7,14)(9,16)(10,15)(11,18)(13,19)(17,20)(21,25)(22,30)(23,33)(24,35)(26,37)(27,39)(28,41)(29,42)(31,44)(32,46)(34,47)(36,48)(38,49)(40,50)(43,51)(45,52)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh134 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534856);