Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499849 |
| 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 | #62886 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(18,23)(21,32)(24,37)(25,36)(26,33)(28,34)(31,43)(35,39)(38,47)(40,44)(42,45)(46,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,18)(19,24)(20,25)(21,33)(22,26)(23,32)(27,38)(28,35)(29,39)(30,40)(31,44)(34,37)(36,43)(41,48)(42,46)(45,47);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,19,22,29)(18,24,26,39)(20,27,30,41)(21,34,32,28)(23,35,33,37)(25,38,40,48)(31,45,43,42)(36,46,44,47);(4,12)(7,16)(9,11)(13,15)(17,20)(18,25)(19,27)(21,31)(22,30)(23,36)(24,38)(26,40)(28,42)(29,41)(32,43)(33,44)(34,45)(35,46)(37,47)(39,48);(3,10)(6,14)(9,11)(13,15)(20,30)(25,40)(27,41)(31,43)(36,44)(38,48)(42,45)(46,47)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6415885);