Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1751 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)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,26,31)(23,27,33,40)(24,29,35,43)(25,28,37,30)(32,39,46,51)(34,38,47,41)(36,42,48,44)(45,49,52,50);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23)(22,27)(24,32)(25,34)(26,33)(28,38)(29,39)(30,41)(31,40)(35,46)(36,45)(37,47)(42,49)(43,51)(44,50)(48,52);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,28)(25,37)(27,38)(29,42)(30,31)(34,47)(36,48)(39,49)(40,41)(43,44)(45,52)(50,51);(6,12)(10,13)(11,18)(17,19)(21,24,26,35)(22,29,31,43)(23,32,33,46)(25,36,37,48)(27,39,40,51)(28,42,30,44)(34,45,47,52)(38,49,41,50);(2,4)(7,15)(10,13)(14,20)(17,19)(23,33)(24,35)(27,40)(29,43)(34,47)(36,48)(38,41)(42,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6577090);