Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1653 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22)(23,31)(24,35)(25,29)(26,39)(27,33)(28,37)(30,38)(32,45)(34,41)(36,50)(40,52)(42,51)(43,47)(44,48)(46,49);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23)(22,27)(24,32)(25,33)(26,34)(28,40)(29,31)(30,41)(35,49)(36,46)(37,47)(38,48)(39,44)(42,43)(45,50)(51,52);(2,4)(21,24,26,37,25,36,38,51)(22,28,30,35,29,42,39,50)(23,32,34,47,33,46,48,52)(27,40,41,49,31,43,44,45);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(22,29)(23,33)(24,36)(30,39)(34,48)(37,51)(40,43)(45,49);(2,4)(7,15)(10,13)(14,20)(18,19)(23,33)(24,36)(27,31)(28,42)(34,48)(35,50)(37,51)(41,44)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591324);