Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1503 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(17,18,22,26)(19,28,30,23)(20,32,34,40)(21,25,35,27)(24,39,41,33)(29,43,44,47)(31,38,46,37)(36,45,42,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,31)(22,30)(24,36)(25,37)(26,28)(27,38)(32,47)(33,45)(34,44)(35,46)(39,48)(40,43)(41,42);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,21,33,22,34,35,39)(18,24,27,32,26,41,25,40)(19,29,31,45,30,44,46,48)(23,36,38,47,28,42,37,43);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(19,30)(23,28)(29,44)(31,46)(36,42)(37,38)(43,47)(45,48);(11,15)(14,16)(17,21,22,35)(18,25,26,27)(19,31,30,46)(20,33,34,39)(23,37,28,38)(24,40,41,32)(29,45,44,48)(36,43,42,47)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6467554);