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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 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)(7,12)(8,10)(9,11)(13,14,20,26)(15,21,28,39)(16,31,35,43)(17,36,27,42)(18,25,19,24)(22,34,44,33)(23,32,38,37)(29,41,30,40);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,32)(17,34)(18,30)(19,29)(20,28)(22,42)(23,43)(24,41)(25,40)(26,39)(27,33)(31,38)(35,37)(36,44);(13,16,20,35)(14,22,26,44)(15,27,28,17)(18,34,19,33)(21,38,39,23)(24,43,25,31)(29,37,30,32)(36,41,42,40);(2,4)(5,8,9,7)(6,10,11,12)(13,17,20,27)(14,23,26,38)(15,16,28,35)(18,32,19,37)(21,22,39,44)(24,42,25,36)(29,34,30,33)(31,40,43,41);(6,11)(10,12)(15,28)(17,27)(21,39)(23,38)(29,30)(32,37)(36,42)(40,41)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=fi,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, 7425286);