Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1746 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)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22)(23,31)(24,35)(25,29)(26,40)(27,33)(28,38)(30,39)(32,46)(34,42)(36,50)(37,49)(41,51)(43,47)(44,52)(45,48);(2,3)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23)(22,27)(24,36)(25,33)(26,34)(28,43)(29,31)(30,42)(32,37)(35,46)(38,51)(39,48)(40,45)(41,44)(47,52)(49,50);(1,2,4,3)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,26,38,25,37,39,52)(22,28,30,35,29,44,40,49)(23,32,34,47,33,36,48,51)(27,41,42,50,31,43,45,46);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(22,29)(23,33)(28,44)(30,40)(32,36)(34,48)(35,49)(47,51);(1,3,4,2)(7,17)(11,14)(15,16)(19,20)Polycyclic presentation:
2^9|c^2=fgh,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722870);