Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | |
1827 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)(15,22)(16,28)(17,33)(18,23)(19,35)(20,26)(21,32)(24,34)(25,38)(27,37)(29,36)(30,40)(31,43)(39,44)(41,42);(1,2)(3,4)(6,9)(10,12)(13,15,18,26)(14,20,23,22)(16,29,31,30)(17,34,27,19)(21,38,39,42)(24,33,35,37)(25,32,41,44)(28,40,43,36);(5,7,8,11)(6,10,9,12)(13,16,19,32,18,31,34,44)(14,21,24,28,23,39,35,43)(15,25,27,30,26,41,17,29)(20,36,37,42,22,40,33,38);(2,4)(13,17,19,15,18,27,34,26)(14,22,24,33,23,20,35,37)(16,30,32,41,31,29,44,25)(21,36,28,42,39,40,43,38);(6,9)(10,12)(14,23)(16,31)(20,22)(24,35)(25,41)(29,30)(32,44)(33,37)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=g,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468399);