Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470231 |
| 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 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3012 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)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(20,30)(23,34)(24,37)(27,41)(31,32)(36,46)(38,39)(42,43);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,35)(28,36)(29,34)(30,37)(31,38)(32,39)(33,40)(41,45)(42,47)(43,48)(44,46);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,35)(25,36)(26,34)(30,41)(31,42)(32,43)(33,44)(37,45)(38,47)(39,48)(40,46);(3,10)(6,14)(9,11)(13,15)(17,20,22,30)(18,24,26,37)(19,27,29,41)(21,32,33,31)(23,35,34,45)(25,39,40,38)(28,43,44,42)(36,48,46,47);(2,8)(6,14)(7,16)(13,15)(17,21,22,33)(18,25,26,40)(19,28,29,44)(20,31,30,32)(23,36,34,46)(24,38,37,39)(27,42,41,43)(35,47,45,48)Polycyclic presentation:
2^9|c^2=h,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,g^b=gi,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, 6384798);