Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,25,22,28,21,27)(19,24,36,38,30,40,31,41)(20,32,37,44,34,26,35,42)(29,43,46,48,33,39,45,47);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,33)(21,31)(22,30)(23,36)(25,38)(26,43)(27,41)(28,40)(29,34)(32,39)(35,46)(37,45)(42,48)(44,47);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(18,25)(21,23)(24,38)(26,42)(27,28)(31,36)(32,44)(35,37)(39,47)(40,41)(43,48)(45,46);(3,8)(6,7)(11,16)(13,15)(17,20)(18,26)(19,29)(21,35)(22,34)(23,37)(24,39)(25,42)(27,44)(28,32)(30,33)(31,45)(36,46)(38,47)(40,43)(41,48);(10,14)(13,15)(19,30)(20,34)(24,40)(26,32)(31,36)(35,37)(38,41)(42,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,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, 6912835);