Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,9,11,12)(13,14)(15,24)(16,21)(17,22)(18,32)(19,23)(20,27)(25,38)(26,34)(28,39)(29,35)(30,37)(31,36)(33,40)(41,43)(42,44);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,27,19,28,17,26)(14,20,22,24,23,34,32,39)(16,25,29,40,30,41,31,42)(21,33,36,38,37,44,35,43);(6,10)(7,11)(13,16,19,30)(14,21,23,37)(15,25,28,41)(17,31,18,29)(20,33,34,44)(22,36,32,35)(24,38,39,43)(26,42,27,40);(2,4)(6,10)(9,12)(13,17,19,18)(14,22,23,32)(15,26,28,27)(16,29,30,31)(20,24,34,39)(21,35,37,36)(25,40,41,42)(33,43,44,38);(13,15,17,26,19,28,18,27)(14,20,32,39,23,34,22,24)(16,25,31,42,30,41,29,40)(21,33,35,43,37,44,36,38)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=i,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434625);