Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,25,30,26,31,38,44)(23,32,35,43,34,28,41,27)(24,29,40,48,39,47,49,52)(33,46,36,50,37,45,51,42);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,34)(22,27,31,43)(24,36,39,51)(25,41,38,35)(28,44,32,30)(29,46,47,45)(33,49,37,40)(42,52,50,48);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,23)(22,28)(24,37)(25,35)(26,34)(27,30)(29,46)(31,32)(33,39)(36,49)(38,41)(40,51)(42,52)(43,44)(45,47)(48,50);(8,16)(11,20)(13,15)(17,19)(21,24,38,49,26,39,25,40)(22,29,44,52,31,47,30,48)(23,33,41,51,34,37,35,36)(27,42,28,45,43,50,32,46);(6,12)(7,14)(11,20)(13,15)(23,34)(24,39)(27,43)(28,32)(29,47)(35,41)(40,49)(48,52)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=gi,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949557);