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 | #42674 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1739 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)(33,43)(34,41)(35,44)(36,46)(37,45)(38,48)(39,47)(49,51)(50,52);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,27)(24,35)(25,32)(26,34)(28,44)(29,41)(30,42)(31,37)(33,39)(36,50)(38,49)(40,45)(43,47)(46,52)(48,51);(1,2,3,4)(21,24,26,36)(22,28,29,46)(23,31,34,49)(25,38,39,37)(27,40,41,51)(30,48,47,45)(32,35,33,50)(42,44,43,52);(6,11)(10,13)(12,19)(18,20)(22,29)(24,36)(27,41)(30,47)(31,49)(35,50)(37,38)(42,43);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,32)(25,39)(27,42)(30,47)(31,35)(33,34)(37,38)(40,44)(41,43)(45,48)(49,50)(51,52)Polycyclic presentation:
2^9|b^2=f,c^2=gi,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, 6593396);