Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2030 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,30,38)(20,33,35,42)(21,27,22,26)(25,36,41,34)(29,43,44,48)(31,40,32,39)(37,46,47,45);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,37)(26,39)(27,40)(28,38)(33,43)(34,45)(35,44)(36,46)(41,47)(42,48);(11,15)(13,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,33)(27,42)(28,41)(30,44)(31,45)(32,46)(38,47)(39,43)(40,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,30)(24,38)(29,44)(31,32)(37,47)(39,40)(43,48)(45,46);(17,21,23,22)(18,26,28,27)(19,31,30,32)(20,34,35,36)(24,39,38,40)(25,33,41,42)(29,45,44,46)(37,43,47,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340647);