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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2682 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)(7,11)(8,10)(9,12)(13,14)(15,24)(16,29)(17,22)(18,32)(19,23)(20,25)(21,30)(26,34)(27,35)(28,38)(31,36)(33,37)(39,41)(40,43)(42,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(15,25)(16,30)(17,18)(20,24)(21,29)(22,32)(27,28)(31,33)(35,38)(36,37)(39,43)(40,41);(13,15,19,28)(14,20,23,35)(16,26,31,42)(17,25,18,27)(21,34,37,44)(22,38,32,24)(29,39,36,43)(30,40,33,41);(2,4)(13,16)(14,21)(15,26)(17,30)(18,33)(19,31)(20,34)(22,36)(23,37)(24,39)(25,40)(27,41)(28,42)(29,32)(35,44)(38,43);(6,10)(9,11)(13,17,19,18)(14,22,23,32)(15,27,28,25)(16,30,31,33)(20,24,35,38)(21,36,37,29)(26,41,42,40)(34,39,44,43)Polycyclic presentation:
2^9|b^2=f,c^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,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, 7421757);