Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,13,16)(11,14,12,15)(17,18,20,22)(19,21,23,24)(25,26)(27,35)(28,31)(29,39)(30,32)(33,37)(34,38)(36,40);(2,5)(6,7)(9,11,13,12)(10,14,16,15)(17,19,20,23)(18,21,22,24)(25,27,29,28,30,37,38,36)(26,31,34,35,32,40,39,33);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,16,14)(19,23)(21,24)(26,32)(27,36)(28,37)(29,38)(31,35)(33,40);(3,8)(6,7)(9,12,13,11)(10,15,16,14)(25,28,38,27,30,36,29,37)(26,33,39,40,32,35,34,31)Polycyclic presentation:
2^9|a^2=efg,b^2=efgh,c^2=fi,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381527);