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)(17,18,22,28,24,29,23,42)(19,30,32,35,33,43,31,25)(20,34,37,45,38,44,36,26)(21,27,39,46,41,47,40,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,27)(21,34)(22,31)(23,32)(24,33)(26,39)(28,43)(29,35)(30,42)(36,46)(37,48)(38,47)(40,45)(41,44);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,23,36,24,38,22,37)(18,26,28,34,29,45,42,44)(19,27,32,46,33,47,31,48)(21,35,40,30,41,25,39,43);(4,12)(7,16)(9,11)(13,15)(17,21,22,39,24,41,23,40)(18,27,28,46,29,47,42,48)(19,26,32,34,33,45,31,44)(20,35,37,43,38,25,36,30);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,28,29,42)(19,31,33,32)(20,36,38,37)(21,39,41,40)(25,43,35,30)(26,44,45,34)(27,46,47,48)Polycyclic presentation:
2^9|a^2=fh,c^2=g,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,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949288);