Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1636 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,31)(24,29)(25,37)(26,28)(27,33)(30,39)(32,45)(34,44)(35,42)(36,49)(38,40)(41,47)(43,48)(46,52)(50,51);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,34)(22,27,28,40)(24,32,35,46)(25,38,39,33)(29,41,42,51)(30,44,37,31)(36,50,48,47)(43,52,49,45);(1,2,3,4)(7,13)(10,17)(14,19)(18,20)(22,28)(27,40)(29,42)(30,37)(31,44)(41,51)(43,49)(45,52);(7,13)(10,17)(14,19)(18,20)(21,24,26,35)(22,29,28,42)(23,32,34,46)(25,36,39,48)(27,41,40,51)(30,43,37,49)(31,45,44,52)(33,47,38,50);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,35)(29,42)(32,46)(36,48)(41,51)(43,49)(45,52)(47,50)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gi,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404759);