Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417692 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1215 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)(10,11)(13,14,17,23,19,24,18,20)(15,25,27,37,29,36,28,21)(16,30,31,22,32,34,33,38)(26,40,41,39,42,35,43,44);(5,6,8,9)(7,11,12,10)(14,20)(16,31)(17,18)(21,25)(22,38)(23,24)(26,41)(27,28)(32,33)(36,37)(39,44)(42,43);(13,15,19,29)(14,21,24,37)(16,26,32,42)(17,28,18,27)(20,25,23,36)(22,35,38,40)(30,44,34,39)(31,43,33,41);(2,4)(5,7,8,12)(6,10,9,11)(13,16,19,32)(14,22,24,38)(15,26,29,42)(17,33,18,31)(20,34,23,30)(21,35,37,40)(25,39,36,44)(27,43,28,41)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337677);