Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499870 |
| 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 | C24 | C25 |
| prime radical | #62820 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3074 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)(17,18)(19,23)(20,24)(21,26)(22,25)(27,39)(28,35)(29,34)(30,37)(31,36)(32,38)(33,41)(40,47)(42,45)(43,46)(44,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,33)(25,34)(26,35)(30,40)(31,41)(32,42)(36,39)(37,44)(38,45)(43,48)(46,47);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20)(18,24)(19,27)(21,30)(22,31)(23,33)(25,36)(26,37)(28,40)(29,41)(32,43)(34,39)(35,44)(38,46)(42,48)(45,47);(4,12)(7,16)(9,11)(13,15)(18,25)(23,34)(24,36)(26,38)(33,39)(35,45)(37,46)(44,47);(3,10)(6,14)(9,11)(13,15)(17,21,22,32)(18,26,25,38)(19,28,29,42)(20,30,31,43)(23,35,34,45)(24,37,36,46)(27,40,41,48)(33,44,39,47)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6415892);