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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1664 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)(17,18,22,29,24,30,23,28)(19,31,34,25,36,48,35,38)(20,37,44,33,40,45,39,26)(21,27,41,47,43,32,42,46);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,24,36)(18,25,30,38)(20,27,40,32)(21,37,43,45)(22,35,23,34)(26,42,33,41)(28,48,29,31)(39,47,44,46);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,22,44,24,40,23,39)(18,26,28,45,30,33,29,37)(19,32,35,47,36,27,34,46)(21,25,41,48,43,38,42,31);(4,12)(7,16)(9,11)(13,15)(17,21,22,41,24,43,23,42)(18,27,29,47,30,32,28,46)(19,33,34,45,36,26,35,37)(20,38,44,31,40,25,39,48);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,28,30,29)(19,34,36,35)(20,39,40,44)(21,41,43,42)(25,31,38,48)(26,37,33,45)(27,46,32,47)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=gi,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,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949302);