Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500006 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62887 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2815 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(18,23,26,32)(24,37,39,35)(25,36,40,44)(38,47,48,46);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,24)(20,25)(21,32)(22,26)(23,34)(27,38)(28,35)(29,39)(30,40)(31,44)(33,37)(36,43)(41,48)(42,46)(45,47);(2,7)(3,9)(4,11)(12,16)(17,19,22,29)(18,24,26,39)(20,27,30,41)(21,33,34,28)(23,35,32,37)(25,38,40,48)(31,45,43,42)(36,46,44,47);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,12,13,16)(17,20)(18,25)(19,27)(21,31)(22,30)(23,36)(24,38)(26,40)(28,42)(29,41)(32,44)(33,45)(34,43)(35,46)(37,47)(39,48);(3,9)(6,13)(10,15)(12,16)(20,30)(25,40)(27,41)(31,43)(36,44)(38,48)(42,45)(46,47)Polycyclic presentation:
2^9|a^2=gh,c^2=i,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6432092);