Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #30397 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1508 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)(3,6)(4,5)(7,8)(17,18,22,27)(19,28,32,38)(20,24,33,40)(21,25,29,26)(23,36,39,31)(30,43,44,48)(34,41,35,42)(37,45,47,46);(1,3,7,4)(2,5,8,6)(9,10)(11,15)(12,14)(13,16)(20,33)(24,40)(30,44)(34,35)(37,47)(41,42)(43,48)(45,46);(1,4,7,3)(2,5,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,29,36,22,32,21,31)(18,23,25,38,27,39,26,28)(20,30,34,45,33,44,35,46)(24,37,42,48,40,47,41,43);(1,5,7,6)(2,3,8,4)(11,16)(13,15)(17,20,22,33)(18,24,27,40)(19,30,32,44)(21,35,29,34)(23,37,39,47)(25,42,26,41)(28,43,38,48)(31,46,36,45);(1,6,7,5)(2,3,8,4)(10,14)(13,15)(17,21,22,29)(18,25,27,26)(19,31,32,36)(20,34,33,35)(23,38,39,28)(24,41,40,42)(30,45,44,46)(37,43,47,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fgi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6823004);