Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(17,18,23,27)(19,24,29,38)(20,31,32,42)(21,26,22,35)(25,36,41,33)(28,43,46,48)(30,39,34,40)(37,44,47,45);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,38)(20,28,32,46)(21,34,22,30)(25,37,41,47)(26,40,35,39)(31,43,42,48)(33,45,36,44);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(19,28)(21,33)(22,36)(23,32)(24,37)(26,42)(27,41)(29,46)(30,44)(31,35)(34,45)(38,47)(39,43)(40,48);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,29)(20,32)(24,38)(25,41)(30,34)(31,42)(33,36)(39,40);(11,15)(14,16)(17,21,23,22)(18,26,27,35)(19,30,29,34)(20,33,32,36)(24,39,38,40)(25,42,41,31)(28,44,46,45)(37,43,47,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6446103);