Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6660408 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35328 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1712 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,29)(19,31,34,25)(20,27,21,36)(22,30,24,28)(26,38,43,35)(32,41,37,44)(33,42,39,45)(40,48,46,47);(1,3,4,8)(2,5,6,7)(11,15)(13,16)(17,19,23,34)(18,25,29,31)(20,32,21,37)(22,39,24,33)(26,40,43,46)(27,44,36,41)(28,45,30,42)(35,48,38,47);(3,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,20)(18,26)(19,32)(21,23)(22,35)(24,38)(25,40)(27,28)(29,43)(30,36)(31,46)(33,47)(34,37)(39,48)(41,42)(44,45);(2,6)(5,7)(11,15)(13,16)(17,21,23,20)(18,27,29,36)(19,32,34,37)(22,35,24,38)(25,41,31,44)(26,30,43,28)(33,48,39,47)(40,45,46,42);(10,14)(13,16)(17,22,23,24)(18,28,29,30)(19,33,34,39)(20,35,21,38)(25,42,31,45)(26,27,43,36)(32,47,37,48)(40,41,46,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6567757);