Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2784 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)(3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(21,22,27,32)(23,33,37,41)(24,30,38,45)(25,31,26,29)(28,36,43,35)(34,47,48,51)(39,46,40,44)(42,50,52,49);(2,4)(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,19,13,18)(21,23,27,37)(22,28,32,43)(24,34,38,48)(25,36,26,35)(29,41,31,33)(30,42,45,52)(39,50,40,49)(44,51,46,47);(5,8,10,9)(6,12,14,13)(7,15,17,16)(11,18,20,19)(22,29)(25,26)(28,41)(30,44)(31,32)(33,43)(35,36)(39,40)(42,51)(45,46)(47,52)(49,50);(5,9,10,8)(6,13,14,12)(7,16,17,15)(11,19,20,18)(21,24,27,38)(22,30,32,45)(23,34,37,48)(25,39,26,40)(28,42,43,52)(29,44,31,46)(33,47,41,51)(35,49,36,50);(6,14)(11,20)(12,13)(18,19)(24,38)(30,45)(34,48)(39,40)(42,52)(44,46)(47,51)(49,50)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396932);