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 | #35634 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,34)(28,36)(29,35)(30,37)(31,40)(32,42)(33,39)(38,44)(41,46)(43,47)(45,48);(2,5)(3,7)(10,13)(11,15)(17,19,22,30)(18,23,26,37)(20,31,32,28)(21,29,34,44)(24,35,27,38)(25,40,42,36)(33,45,46,43)(39,48,41,47);(9,11,12,15)(10,14,13,16)(18,24)(21,34)(23,35)(25,39)(26,27)(29,44)(33,46)(36,47)(37,38)(40,48)(41,42)(43,45);(2,5)(6,8)(10,13)(14,16)(17,20,22,32)(18,25,26,42)(19,28,30,31)(21,33,34,46)(23,36,37,40)(24,39,27,41)(29,43,44,45)(35,47,38,48);(1,3,4,7)(2,6,5,8)(18,26)(20,32)(23,37)(24,27)(28,31)(33,46)(35,38)(43,45)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=fh,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, 6755331);