Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1730 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)(17,18,22,26)(19,27,30,41)(20,24,32,38)(21,34,28,25)(23,35,37,31)(29,40,44,48)(33,45,42,39)(36,46,47,43);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,28,35,22,30,21,31)(18,23,34,41,26,37,25,27)(20,29,42,46,32,44,33,43)(24,36,45,48,38,47,39,40);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,24)(19,29)(21,33)(22,32)(23,36)(25,39)(26,38)(27,40)(28,42)(30,44)(31,43)(34,45)(35,46)(37,47)(41,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(19,30)(20,32)(23,37)(24,38)(27,41)(31,35)(33,42)(39,45)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6938943);