Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22,26,30)(23,31,34,44)(24,28,35,42)(25,37,39,29)(27,38,41,33)(32,45,47,51)(36,50,49,43)(40,46,52,48);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,27)(24,32)(25,38)(26,34)(28,40)(29,44)(30,41)(31,37)(33,39)(35,47)(36,46)(42,52)(43,51)(45,50)(48,49);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,26,35)(22,28,30,42)(23,32,34,47)(25,36,39,49)(27,40,41,52)(29,43,37,50)(31,45,44,51)(33,48,38,46);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,33)(24,35)(25,39)(27,31)(28,42)(29,37)(32,46)(34,38)(40,51)(41,44)(45,52)(47,48);(6,12)(10,13)(11,20)(18,19)(24,35)(28,42)(32,47)(36,49)(40,52)(43,50)(45,51)(46,48)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7031771);