Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454426 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1124 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)(6,10)(7,8)(13,14,18,24,20,26,19,25)(15,21,28,22,16,30,29,37)(17,32,33,23,35,39,34,40)(27,38,41,36,31,44,42,43);(5,6,9,10)(7,11,8,12)(13,15,18,28,20,16,19,29)(14,21,24,22,26,30,25,37)(17,27,34,42,35,31,33,41)(23,36,32,38,40,43,39,44);(2,4)(5,7)(6,11)(8,9)(10,12)(13,16,20,15)(14,22,26,37)(17,31,35,27)(18,28,19,29)(21,25,30,24)(23,38,40,44)(32,36,39,43)(33,41,34,42);(5,8,9,7)(6,12,10,11)(13,17)(14,23)(15,27)(16,31)(18,34)(19,33)(20,35)(21,36)(22,38)(24,32)(25,39)(26,40)(28,42)(29,41)(30,43)(37,44)Polycyclic presentation:
2^9|a^2=ef,b^2=egi,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, 372019);