Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,23,19,22,18,24)(15,25,27,35,29,36,28,20)(16,21,31,39,33,38,32,40)(26,41,42,44,43,37,30,34);(2,4)(13,15,19,29)(14,20,22,35)(16,30,33,42)(17,28,18,27)(21,37,38,41)(23,36,24,25)(26,31,43,32)(34,39,44,40);(6,9)(10,12)(13,16,17,31,19,33,18,32)(14,21,23,39,22,38,24,40)(15,26,27,42,29,43,28,30)(20,34,25,41,35,44,36,37);(13,16,17,31,19,33,18,32)(14,21,23,39,22,38,24,40)(15,26,27,42,29,43,28,30)(20,34,25,41,35,44,36,37);(5,7,8,11)(6,10,9,12)(14,22)(20,35)(21,38)(23,24)(25,36)(34,44)(37,41)(39,40)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374675);