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×#3269 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 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)(13,14,18,23)(15,19,26,35)(16,28,30,40)(17,21,33,22)(20,32,38,31)(24,37,41,44)(25,36,27,39)(29,34,42,43);(1,2)(3,4)(6,9)(10,12)(13,15,18,26)(14,19,23,35)(16,29,30,42)(17,25,33,27)(20,37,38,44)(21,36,22,39)(24,32,41,31)(28,34,40,43);(13,16,18,30)(14,20,23,38)(15,24,26,41)(17,32,33,31)(19,34,35,43)(21,28,22,40)(25,29,27,42)(36,44,39,37);(2,4)(5,7,8,11)(6,10,9,12)(14,21)(15,25)(17,33)(20,28)(22,23)(24,29)(26,27)(31,32)(36,39)(37,44)(38,40)(41,42);(6,9)(10,12)(15,26)(16,30)(19,35)(20,38)(25,27)(28,40)(31,32)(36,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=f,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,e^c=ei,g^c=gi,h^c=hi,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, 7423980);