Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,23,29)(19,30,34,25)(20,35,33,26)(21,39,24,27)(22,28,31,42)(32,43,41,44)(36,45,40,48)(37,46,38,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,34)(18,25,29,30)(20,36,33,40)(21,32,24,41)(22,38,31,37)(26,45,35,48)(27,43,39,44)(28,46,42,47);(10,14)(13,15)(17,20,23,33)(18,26,29,35)(19,31,34,22)(21,38,24,37)(25,42,30,28)(27,47,39,46)(32,40,41,36)(43,45,44,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,24)(18,27,29,39)(19,32,34,41)(20,37,33,38)(22,40,31,36)(25,43,30,44)(26,46,35,47)(28,45,42,48);(2,5)(6,8)(11,16)(13,15)(17,22,23,31)(18,28,29,42)(19,33,34,20)(21,40,24,36)(25,35,30,26)(27,45,39,48)(32,38,41,37)(43,47,44,46)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6606335);