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 | #61420 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18,22,26,23,20,34,27)(19,28,31,29,33,43,32,24)(21,25,38,41,37,35,45,42)(30,40,36,44,46,48,47,39);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19,23,33)(18,24,20,29)(21,36,37,47)(22,32,34,31)(25,40,35,48)(26,43,27,28)(30,45,46,38)(39,42,44,41);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,23)(19,29)(21,35)(22,27)(24,33)(25,37)(26,34)(28,32)(30,44)(31,43)(36,48)(38,42)(39,46)(40,47)(41,45);(3,10)(6,9)(11,16)(14,15)(17,21,34,45,23,37,22,38)(18,25,27,42,20,35,26,41)(19,30,32,47,33,46,31,36)(24,39,43,48,29,44,28,40);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(21,37)(25,35)(30,46)(36,47)(38,45)(39,44)(40,48)(41,42)Polycyclic presentation:
2^9|a^2=f,b^2=hi,d^2=fi,e^2=h,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, 7088507);