Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1693 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)(8,16)(11,17)(15,20)(18,19)(21,22,25,31,26,30,37,28)(23,27,34,38,32,40,46,41)(24,29,36,43,35,44,48,42)(33,39,47,49,45,51,52,50);(1,2,3,4)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,27)(24,33)(25,34)(26,32)(28,41)(29,39)(30,40)(31,38)(35,45)(36,47)(37,46)(42,50)(43,49)(44,51)(48,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(22,28)(23,32)(24,35)(25,37)(27,38)(29,43)(30,31)(39,50)(40,41)(42,44)(47,52)(49,51);(2,4)(7,14)(10,13)(15,20)(18,19)(21,24,26,35)(22,29,30,44)(23,33,32,45)(25,36,37,48)(27,39,40,51)(28,42,31,43)(34,47,46,52)(38,49,41,50);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,30)(23,32)(24,35)(28,31)(34,46)(36,48)(39,51)(49,50)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089696);