Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628847 |
| 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×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,23,32)(16,26,28,36)(17,29,31,22)(20,30,35,27)(24,39,37,44)(25,41,38,34)(33,42,43,40);(6,9)(10,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,33)(21,32)(22,34)(26,39)(27,40)(28,37)(29,41)(30,42)(31,38)(35,43)(36,44);(1,2,3,4)(15,23)(19,32)(24,37)(25,38)(33,43)(34,41)(39,44)(40,42);(13,16,18,28)(14,20,21,35)(15,24,23,37)(17,30,31,27)(19,33,32,43)(22,36,29,26)(25,42,38,40)(34,44,41,39);(5,7,8,11)(6,10,9,12)(14,21)(19,32)(20,35)(22,29)(26,36)(33,43)(34,41)(39,44)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,h^a=hi,c^b=ci,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537406);