Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 | C2×#3561 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(13,14)(15,25)(16,30)(17,22)(18,24)(19,23)(20,28)(21,32)(26,40)(27,35)(29,36)(31,37)(33,38)(34,43)(39,42)(41,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,23,35)(16,26,33,43)(17,28,18,27)(21,34,37,40)(22,36,24,25)(30,44,38,39)(31,42,32,41);(6,9)(10,12)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,30)(23,37)(24,38)(25,39)(27,41)(28,42)(29,43)(35,40)(36,44);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,24)(15,27,29,28)(16,31,33,32)(20,25,35,36)(21,30,37,38)(26,41,43,42)(34,39,40,44);(7,11)(10,12)(14,23)(20,35)(21,37)(22,24)(25,36)(30,38)(34,40)(39,44)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449943);