Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14)(15,25)(16,29)(17,22)(18,35)(19,24)(20,27)(21,26)(23,30)(28,42)(31,38)(32,37)(33,36)(34,41)(39,43)(40,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,27,19,28,30,31)(14,20,23,25,24,38,35,42)(16,17,26,34,32,33,43,44)(21,36,29,41,39,22,37,40);(6,9)(10,12)(13,16,30,43,19,32,18,26)(14,21,35,37,24,39,23,29)(15,17,31,44,28,33,27,34)(20,36,42,40,38,22,25,41);(6,9)(10,12)(13,17)(14,22)(15,26)(16,31)(18,34)(19,33)(20,37)(21,25)(23,40)(24,36)(27,32)(28,43)(29,38)(30,44)(35,41)(39,42);(5,7,8,11)(6,10,9,12)(16,32)(17,33)(21,39)(22,36)(26,43)(29,37)(34,44)(40,41)Polycyclic presentation:
2^9|b^2=f,c^2=gi,e^2=h,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,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6401607);