Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1949 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)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,23,18)(16,25,30,37)(17,26,34,40)(21,29,27,31)(22,35,39,33)(28,36,32,38)(41,43,42,44);(1,2)(3,4)(6,9)(10,12)(13,15,19,23)(14,18,24,20)(16,27,30,21)(17,22,34,39)(25,29,37,31)(26,33,40,35)(28,42,32,41)(36,43,38,44);(5,7,8,11)(6,10,9,12)(13,16,19,30)(14,21,24,27)(15,25,23,37)(17,32,34,28)(18,29,20,31)(22,38,39,36)(26,42,40,41)(33,44,35,43);(2,4)(13,17,19,34)(14,22,24,39)(15,26,23,40)(16,28,30,32)(18,35,20,33)(21,36,27,38)(25,41,37,42)(29,44,31,43);(6,9)(10,12)(13,18,19,20)(14,23,24,15)(16,29,30,31)(17,33,34,35)(21,37,27,25)(22,26,39,40)(28,43,32,44)(36,41,38,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422563);