Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 | #61406 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1662 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)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,22,26,23,28,34,27)(19,29,32,36,33,20,31,24)(21,25,38,42,39,44,47,43)(30,41,37,48,46,35,45,40);(3,9)(4,12)(6,13)(7,11)(17,19)(18,24)(20,26)(21,37)(22,31)(23,33)(25,41)(27,29)(28,36)(30,38)(32,34)(35,44)(39,45)(40,42)(43,48)(46,47);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,20,34,36,23,29,22,24)(18,19,26,32,28,33,27,31)(21,35,47,48,39,41,38,40)(25,30,42,37,44,46,43,45);(3,9)(6,13)(10,15)(14,16)(17,21,34,47,23,39,22,38)(18,25,27,43,28,44,26,42)(19,30,31,45,33,46,32,37)(20,35,36,48,29,41,24,40);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,22,23,34)(18,26,28,27)(19,31,33,32)(20,36,29,24)(21,38,39,47)(25,42,44,43)(30,45,46,37)(35,48,41,40)Polycyclic presentation:
2^9|a^2=fh,c^2=ghi,d^2=fi,e^2=hi,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,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104020);