Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #23975 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1844 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)(15,19)(16,28)(17,20)(18,23)(21,31)(22,32)(24,35)(25,37)(26,39)(27,36)(29,34)(30,40)(33,38)(41,44)(42,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,27)(14,19,23,36)(16,29,30,42)(17,24,32,26)(20,35,22,39)(21,37,38,44)(25,33,41,31)(28,34,40,43);(5,8,9,7)(6,10,11,12)(14,20)(15,24)(17,32)(21,28)(22,23)(25,29)(26,27)(31,33)(35,39)(37,44)(38,40)(41,42);(2,4)(13,16)(14,21)(15,25)(17,33)(18,30)(19,34)(20,28)(22,40)(23,38)(24,29)(26,42)(27,41)(31,32)(35,44)(36,43)(37,39);(6,11)(10,12)(16,30)(21,38)(25,41)(28,40)(29,42)(31,33)(34,43)(37,44)Polycyclic presentation:
2^9|b^2=fi,c^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,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422712);