Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1949 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)(5,6,8,10)(7,11,12,9)(15,24)(16,28)(17,18)(20,25)(21,36)(22,30)(27,31)(29,32)(33,43)(34,35)(37,38)(39,41);(5,7,8,12)(6,9,10,11)(13,14)(15,25)(16,21)(17,30)(18,22)(19,23)(20,24)(26,40)(27,34)(28,36)(29,38)(31,35)(32,37)(33,39)(41,43)(42,44);(2,4)(13,15)(14,20)(16,26)(17,31)(18,24)(19,27)(21,33)(22,34)(23,35)(25,30)(28,41)(29,42)(32,39)(36,40)(37,44)(38,43);(7,12)(9,11)(13,16,19,29)(14,21,23,38)(15,26,27,42)(17,28,18,32)(20,33,35,43)(22,37,30,36)(24,39,31,41)(25,40,34,44)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325173);