Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #29607 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1508 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)(17,18,21,26)(19,28,30,23)(20,32,34,25)(22,37,36,27)(24,38,43,35)(29,40,33,44)(31,41,47,45)(39,46,42,48);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,33)(21,30)(22,31)(24,42)(25,44)(26,28)(27,41)(29,34)(32,40)(35,48)(36,47)(37,45)(38,46)(39,43);(1,3,5,8)(2,7,6,4)(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,29)(21,34)(22,38)(23,39)(25,27)(26,43)(28,42)(30,33)(31,48)(32,37)(35,36)(40,41)(44,45)(46,47);(2,6)(3,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,40)(24,27)(26,32)(28,44)(30,33)(31,46)(36,38)(37,43)(39,41)(42,45)(47,48);(1,4,5,7)(2,8,6,3)(10,14)(13,15)(19,30)(20,34)(23,28)(24,43)(25,32)(31,47)(35,38)(41,45)Polycyclic presentation:
2^9|a^2=i,c^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6797882);