Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,12)(8,10)(9,11)(13,14)(15,20)(16,27)(17,22)(18,33)(19,23)(21,29)(24,41)(25,35)(26,37)(28,44)(30,39)(31,36)(32,40)(34,43)(38,42);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,26)(14,20,23,37)(16,28,30,24)(17,31,18,25)(21,38,40,34)(22,36,33,35)(27,44,39,41)(29,42,32,43);(2,4)(13,16,18,29,19,30,17,32)(14,21,22,27,23,40,33,39)(15,24,25,43,26,28,31,42)(20,34,36,41,37,38,35,44);(2,4)(5,8,9,7)(6,10,11,12)(13,15)(14,20)(16,24)(17,31)(18,25)(19,26)(21,34)(22,36)(23,37)(27,41)(28,30)(29,43)(32,42)(33,35)(38,40)(39,44);(13,17,19,18)(14,22,23,33)(15,25,26,31)(16,29,30,32)(20,35,37,36)(21,39,40,27)(24,42,28,43)(34,41,38,44)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351141);