Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | |
1780 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)(6,8)(11,15)(14,16)(17,18,22,27)(19,23,30,38)(20,31,32,25)(21,24,34,26)(28,41,42,36)(29,35,44,37)(33,40,45,39)(43,47,48,46);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19,22,30)(18,23,27,38)(20,28,32,42)(21,29,34,44)(24,35,26,37)(25,36,31,41)(33,43,45,48)(39,46,40,47);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(18,24)(21,34)(23,35)(25,39)(26,27)(29,44)(31,40)(33,45)(36,46)(37,38)(41,47)(43,48);(2,5)(6,8)(10,13)(14,16)(17,20)(18,25)(19,28)(21,33)(22,32)(23,36)(24,39)(26,40)(27,31)(29,43)(30,42)(34,45)(35,46)(37,47)(38,41)(44,48);(9,11,12,15)(10,14,13,16)(20,32)(25,31)(28,42)(33,45)(36,41)(39,40)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7051531);