Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,32)(24,36)(25,30)(26,28)(27,34)(29,37)(31,39)(33,41)(35,42)(38,44)(40,48)(43,49)(45,46)(47,50)(51,52);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,27)(24,33)(25,34)(26,35)(28,40)(29,41)(30,32)(31,42)(36,45)(37,46)(38,47)(39,48)(43,50)(44,51)(49,52);(2,4)(22,28)(26,39)(27,40)(29,43)(30,31)(32,42)(35,48)(36,44)(38,49)(41,50)(45,51)(47,52);(6,11)(10,13)(12,19)(18,20)(21,24,25,37)(22,29,30,36)(23,33,34,46)(26,38,39,49)(27,41,32,45)(28,43,31,44)(35,47,48,52)(40,50,42,51);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,34)(24,37)(27,32)(29,36)(35,48)(38,49)(40,42)(43,44)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580939);