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 | #35757 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2686 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)(13,14,20,25)(15,21,27,40)(16,28,32,44)(17,33,34,42)(18,24,19,35)(22,30,41,31)(23,37,43,29)(26,38,36,39);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,29)(17,31)(18,26)(19,36)(20,27)(22,33)(23,28)(24,38)(25,40)(30,34)(32,37)(35,39)(41,42)(43,44);(13,16,20,32)(14,22,25,41)(15,17,27,34)(18,31,19,30)(21,23,40,43)(24,44,35,28)(26,37,36,29)(33,39,42,38);(2,4)(13,17)(14,23)(15,16)(18,29)(19,37)(20,34)(21,22)(24,42)(25,43)(26,30)(27,32)(28,38)(31,36)(33,35)(39,44)(40,41);(6,10)(9,11)(13,18,20,19)(14,24,25,35)(15,26,27,36)(16,30,32,31)(17,29,34,37)(21,38,40,39)(22,28,41,44)(23,42,43,33)Polycyclic presentation:
2^9|a^2=i,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,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,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, 7421901);