Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1724 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,26,30)(23,27,34,42)(24,35,36,28)(25,29,38,44)(31,39,33,41)(32,46,48,40)(37,49,50,43)(45,52,47,51);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,27)(24,32)(25,31)(26,34)(28,40)(29,39)(30,42)(33,38)(35,46)(36,48)(37,45)(41,44)(43,51)(47,50)(49,52);(1,2,3,4)(23,31)(24,36)(25,38)(27,39)(28,35)(29,44)(32,47)(33,34)(40,52)(41,42)(45,48)(46,51);(6,11)(10,13)(12,19)(18,20)(21,24,26,36)(22,28,30,35)(23,32,34,48)(25,37,38,50)(27,40,42,46)(29,43,44,49)(31,45,33,47)(39,51,41,52);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,31)(25,38)(27,39)(29,44)(32,45)(33,34)(37,50)(40,51)(41,42)(43,49)(46,52)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593410);