Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22)(23,29)(24,39)(25,31)(26,32)(27,42)(28,33)(30,40)(34,48)(35,44)(36,45)(37,51)(38,46)(41,47)(43,49)(50,52);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,28,38)(22,29,33,46)(24,34,25,35)(26,36,27,37)(30,43,47,52)(31,44,39,48)(32,45,42,51)(40,49,41,50);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,30)(23,34)(25,28)(26,40)(27,41)(29,43)(31,32)(33,47)(35,38)(36,49)(37,50)(39,42)(44,45)(46,52)(48,51);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,25,28,24)(22,31,33,39)(23,35,38,34)(26,40,27,41)(29,44,46,48)(30,42,47,32)(36,49,37,50)(43,51,52,45);(6,12)(7,14)(11,20)(13,15)(21,26,28,27)(22,32,33,42)(23,36,38,37)(24,40,25,41)(29,45,46,51)(30,31,47,39)(34,49,35,50)(43,44,52,48)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=h,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 7011722);