Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1444 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)(17,18,23,28)(19,29,32,42)(20,25,33,39)(21,27,22,26)(24,36,38,31)(30,43,44,48)(34,41,35,40)(37,45,47,46);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(20,33)(25,39)(30,44)(34,35)(37,47)(40,41)(43,48)(45,46);(3,8)(5,7)(10,13)(14,15)(17,19,23,32)(18,24,28,38)(20,30,33,44)(21,31,22,36)(25,37,39,47)(26,29,27,42)(34,45,35,46)(40,48,41,43);(2,6)(5,7)(17,20)(18,25)(19,30)(21,35)(22,34)(23,33)(24,37)(26,41)(27,40)(28,39)(29,43)(31,46)(32,44)(36,45)(38,47)(42,48);(9,11,12,16)(10,14,13,15)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,33,35)(24,29,38,42)(25,40,39,41)(30,45,44,46)(37,48,47,43)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6707529);