Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #45388 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,19,25)(15,26,29,21)(16,30,34,22)(17,35,28,23)(18,37,20,24)(27,39,31,40)(32,41,36,44)(33,42,38,43);(2,3)(6,9)(10,11)(13,15,19,29)(14,21,25,26)(16,31,34,27)(17,18,28,20)(22,40,30,39)(23,37,35,24)(32,38,36,33)(41,42,44,43);(1,2,4,3)(13,16,19,34)(14,22,25,30)(15,27,29,31)(17,36,28,32)(18,38,20,33)(21,39,26,40)(23,44,35,41)(24,43,37,42);(1,3,4,2)(13,17,20,15,19,28,18,29)(14,23,24,21,25,35,37,26)(16,32,33,31,34,36,38,27)(22,41,43,40,30,44,42,39);(5,7,8,12)(6,10,9,11)(13,18,19,20)(14,24,25,37)(15,28,29,17)(16,33,34,38)(21,23,26,35)(22,42,30,43)(27,36,31,32)(39,41,40,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521040);