Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61266 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,23,28,24,20,22,27)(19,29,32,30,34,44,33,25)(21,26,37,42,38,35,39,43)(31,41,36,45,47,48,46,40);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,24,34)(18,25,20,30)(21,36,38,46)(22,32,23,33)(26,41,35,48)(27,29,28,44)(31,39,47,37)(40,43,45,42);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20)(18,24)(19,30)(21,35)(22,28)(23,27)(25,34)(26,38)(29,33)(31,45)(32,44)(36,48)(37,43)(39,42)(40,47)(41,46);(4,12)(7,16)(9,11)(13,15)(17,21,22,39,24,38,23,37)(18,26,27,43,20,35,28,42)(19,31,33,46,34,47,32,36)(25,40,44,48,30,45,29,41);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,27,20,28)(19,32,34,33)(21,37,38,39)(25,29,30,44)(26,42,35,43)(31,46,47,36)(40,48,45,41)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=h,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7012010);