Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,11,8,10)(13,14,18,22)(15,24,26,20)(16,28,32,31)(17,30,23,19)(21,37,39,33)(25,42,43,38)(27,34,35,40)(29,41,44,36);(2,4)(5,7,9,8)(6,10,12,11)(13,15,18,26)(14,20,22,24)(16,29,32,44)(17,34,23,40)(19,35,30,27)(21,38,39,42)(25,37,43,33)(28,36,31,41);(13,16,30,37,18,32,19,33)(14,21,17,31,22,39,23,28)(15,25,35,44,26,43,27,29)(20,36,40,42,24,41,34,38);(1,2,3,4)(13,17,19,14,18,23,30,22)(15,20,27,34,26,24,35,40)(16,31,33,21,32,28,37,39)(25,36,29,38,43,41,44,42);(5,8,9,7)(6,11,12,10)(14,22)(17,23)(20,24)(21,39)(28,31)(34,40)(36,41)(38,42)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=gi,d^2=h,e^2=fi,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,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521377);