Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| 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 | #29736 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1534 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,32)(21,25)(22,27)(24,36)(26,33)(28,45)(29,39)(30,43)(31,40)(34,48)(35,42)(37,44)(38,46)(41,47);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,23,27,40)(20,28,37,34)(21,30,33,29)(24,38,42,41)(25,43,26,39)(32,45,44,48)(35,47,36,46);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,33,35,22,37,21,36)(18,24,25,44,27,42,26,32)(19,28,29,47,31,34,30,46)(23,38,43,48,40,41,39,45);(1,4,5,3)(2,8,6,7)(11,15)(13,16)(17,19)(18,23)(20,34)(21,30)(22,31)(24,41)(25,43)(26,39)(27,40)(28,37)(29,33)(32,48)(35,46)(36,47)(38,42)(44,45);(10,14)(13,16)(17,21,22,33)(18,25,27,26)(19,29,31,30)(20,35,37,36)(23,39,40,43)(24,32,42,44)(28,46,34,47)(38,48,41,45)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6381146);