Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(6,9)(10,12)(13,14,19,24,18,23,26,35)(15,25,28,21,16,20,30,37)(17,31,33,40,32,41,34,22)(27,39,43,36,29,38,44,42);(5,6,8,9)(7,10,11,12)(13,15,26,30,18,16,19,28)(14,20,35,21,23,25,24,37)(17,27,33,43,32,29,34,44)(22,36,31,38,40,42,41,39);(2,4)(7,11)(10,12)(13,16,18,15)(14,21,23,37)(17,27,32,29)(19,28,26,30)(20,24,25,35)(22,39,40,38)(31,42,41,36)(33,44,34,43);(5,7,8,11)(6,10,9,12)(13,17,18,32)(14,22,23,40)(15,27,16,29)(19,34,26,33)(20,36,25,42)(21,38,37,39)(24,41,35,31)(28,44,30,43)Polycyclic presentation:
2^9|a^2=f,b^2=eg,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371946);