Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6272 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)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(25,38)(29,42)(33,47)(36,49)(40,44)(41,48)(45,52)(50,51);(1,2)(3,4)(21,22,26,30)(23,31,34,27)(24,28,37,35)(25,29,38,42)(32,43,46,39)(33,44,47,40)(36,41,49,48)(45,51,52,50);(6,10)(11,13)(12,17)(18,19)(21,22)(23,27)(24,35)(25,29)(26,30)(28,37)(31,34)(32,43)(33,40)(36,48)(38,42)(39,46)(41,49)(44,47)(45,51)(50,52);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,23,26,34)(22,27,30,31)(24,32,37,46)(25,33,38,47)(28,39,35,43)(29,40,42,44)(36,45,49,52)(41,50,48,51);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24)(22,28)(23,32)(25,36)(26,37)(27,39)(29,41)(30,35)(31,43)(33,45)(34,46)(38,49)(40,50)(42,48)(44,51)(47,52);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,25)(22,29)(23,33)(24,36)(26,38)(27,40)(28,41)(30,42)(31,44)(32,45)(34,47)(35,48)(37,49)(39,50)(43,51)(46,52)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7621675);