Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3204 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2108 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)(23,31)(24,29)(25,38)(26,28)(27,34)(30,32)(33,44)(35,43)(36,41)(37,49)(39,45)(40,48)(42,46)(47,51)(50,52);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,32,45,26,35,25,34)(22,27,38,43,28,39,30,31)(24,33,46,52,36,47,37,48)(29,40,49,51,41,50,42,44);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(22,28)(24,36)(27,39)(30,38)(31,43)(33,47)(37,46)(48,52);(8,16)(11,20)(13,15)(17,19)(21,24)(22,29)(23,33)(25,37)(26,36)(27,40)(28,41)(30,42)(31,44)(32,46)(34,48)(35,47)(38,49)(39,50)(43,51)(45,52);(7,14)(10,18)(13,15)(17,19)(24,36)(29,41)(33,47)(37,46)(40,50)(42,49)(44,51)(48,52)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6418081);