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 | #45332 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2005 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)(7,9)(8,10)(11,12)(13,14,19,24)(15,20,27,35)(16,21,31,39)(17,22,18,23)(25,33,41,43)(26,34,32,40)(28,36,42,44)(29,37,30,38);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,28)(17,26)(18,32)(19,27)(21,36)(22,34)(23,40)(24,35)(25,30)(29,41)(31,42)(33,38)(37,43)(39,44);(6,10)(7,11)(13,15)(14,20)(16,25)(17,32)(18,26)(19,27)(21,33)(22,40)(23,34)(24,35)(28,30)(29,42)(31,41)(36,38)(37,44)(39,43);(6,10)(9,12)(13,16,17,29,19,31,18,30)(14,21,22,37,24,39,23,38)(15,25,32,42,27,41,26,28)(20,33,40,44,35,43,34,36);(1,3,4,2)(13,17,19,18)(14,22,24,23)(15,26,27,32)(16,29,31,30)(20,34,35,40)(21,37,39,38)(25,28,41,42)(33,36,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7519108);