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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1615 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)(19,31,33,25,20,35,34,26)(21,27,40,44,39,46,38,45)(32,43,47,41,36,48,37,42);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,24)(21,37)(22,34)(23,33)(26,30)(27,43)(28,35)(29,31)(32,38)(36,40)(39,47)(41,45)(42,44)(46,48);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20,24,19)(18,26,30,25)(21,32,39,36)(22,34,23,33)(27,41,46,42)(28,35,29,31)(37,40,47,38)(43,44,48,45);(4,12)(7,16)(9,11)(13,15)(17,21,23,38,24,39,22,40)(18,27,28,45,30,46,29,44)(19,32,34,37,20,36,33,47)(25,41,31,43,26,42,35,48);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,28,30,29)(19,33,20,34)(21,38,39,40)(25,31,26,35)(27,44,46,45)(32,37,36,47)(41,48,42,43)Polycyclic presentation:
2^9|a^2=fh,c^2=h,d^2=f,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, 7012000);