Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #36015 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1737 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)(23,27)(24,28)(25,30)(26,29)(31,40)(32,42)(33,47)(34,41)(35,43)(36,44)(37,45)(38,48)(39,46)(49,51)(50,52);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,26,32)(22,27,29,42)(24,35,36,50)(25,33,39,34)(28,43,44,52)(30,47,46,41)(31,38,49,37)(40,48,51,45);(1,2,3,4)(21,24,26,36)(22,28,29,44)(23,31,32,49)(25,38,39,37)(27,40,42,51)(30,48,46,45)(33,35,34,50)(41,52,47,43);(6,11)(10,13)(12,19)(18,20)(22,29)(23,32)(24,36)(30,46)(33,34)(37,38)(40,51)(43,52);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,29)(23,33)(25,39)(27,41)(28,44)(31,35)(32,34)(37,38)(40,52)(42,47)(43,51)(49,50)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,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, 6592990);