Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499804 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23941 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2838 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)(21,22)(23,32)(24,29)(25,30)(26,39)(27,31)(28,34)(33,47)(35,42)(36,44)(37,45)(38,46)(40,43)(41,50)(48,52)(49,51);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,28)(24,33)(25,34)(26,40)(27,35)(29,41)(30,42)(31,43)(32,39)(36,48)(37,49)(38,50)(44,51)(45,47)(46,52);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,29)(23,33)(25,38)(26,37)(27,36)(28,41)(30,46)(31,44)(32,47)(34,50)(35,48)(39,45)(40,49)(42,52)(43,51);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(24,36)(29,44)(33,48)(37,38)(41,51)(45,46)(47,52)(49,50);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,31,39)(23,34,35,40)(24,37,36,38)(28,42,43,32)(29,45,44,46)(33,49,48,50)(41,47,51,52)Polycyclic presentation:
2^9|c^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436176);