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 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
2916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
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)(17,18,23,22)(19,24,30,29)(20,25,33,32)(21,34,35,26)(27,39,41,36)(28,42,43,37)(31,44,45,38)(40,46,48,47);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,36)(26,37)(31,40)(32,39)(33,41)(34,42)(35,43)(38,46)(44,47)(45,48);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20)(18,25)(19,27)(21,31)(22,32)(23,33)(24,36)(26,38)(28,40)(29,39)(30,41)(34,44)(35,45)(37,46)(42,47)(43,48);(3,10)(6,14)(9,11)(13,15)(17,21,23,35)(18,26,22,34)(19,28,30,43)(20,31,33,45)(24,37,29,42)(25,38,32,44)(27,40,41,48)(36,46,39,47);(2,8)(6,14)(7,16)(13,15)(17,22,23,18)(19,29,30,24)(20,32,33,25)(21,26,35,34)(27,39,41,36)(28,37,43,42)(31,38,45,44)(40,46,48,47)Polycyclic presentation:
2^9|a^2=i,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, 6384799);