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 |
| prime radical | C2×#3535 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 52 points:
(2,4)(6,11)(7,12)(9,17)(15,20)(25,37)(30,39)(33,44)(35,48)(41,49)(42,47)(46,51)(50,52);(1,2)(3,4)(5,6,10,11)(7,8,12,16)(9,13,17,18)(14,15,19,20)(22,27)(23,31)(29,34)(30,39)(32,43)(33,44)(42,47)(46,51);(5,7)(6,8)(9,14)(10,12)(11,16)(13,15)(17,19)(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);(5,8)(6,12)(7,11)(9,15)(10,16)(13,19)(14,18)(17,20)(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);(5,9)(6,13)(7,14)(8,15)(10,17)(11,18)(12,19)(16,20)(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);(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|b^2=h,c^2=i,e^2=i,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, 7621937);