Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(2,4)(5,6,9,12)(7,10,8,11)(13,14,19,24)(15,21,16,30)(17,32,35,42)(18,25,26,23)(20,29,38,28)(22,36,41,34)(27,37,31,44)(33,39,43,40);(1,2,3,4)(5,7,9,8)(6,10,12,11)(13,15,26,29,19,16,18,28)(14,20,25,30,24,38,23,21)(17,33,36,27,35,43,34,31)(22,40,42,37,41,39,32,44);(5,7,9,8)(6,11,12,10)(13,16,19,15)(14,21,24,30)(17,31,35,27)(18,29,26,28)(20,23,38,25)(22,39,41,40)(32,37,42,44)(33,34,43,36);(2,4)(5,8,9,7)(6,11,12,10)(13,17)(14,22)(15,27)(16,31)(18,36)(19,35)(20,37)(21,39)(23,42)(24,41)(25,32)(26,34)(28,43)(29,33)(30,40)(38,44)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339440);