Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,28)(20,32)(21,27)(22,24)(23,36)(25,30)(26,33)(29,44)(31,40)(34,42)(35,41)(37,38)(39,46)(43,45)(47,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(20,33)(25,38)(27,36)(28,40)(29,45)(34,35)(46,47);(9,11,12,16)(10,13,14,15)(17,19,22,31)(18,25,24,38)(20,29,33,45)(21,30,23,37)(26,39,32,48)(27,40,36,28)(34,46,35,47)(41,43,42,44);(3,8)(5,7)(10,14)(11,16)(17,20)(18,26)(19,29)(21,35)(22,33)(23,34)(24,32)(25,39)(27,42)(28,43)(30,47)(31,45)(36,41)(37,46)(38,48)(40,44);(2,6)(5,7)(10,14)(13,15)(17,21,22,23)(18,27,24,36)(19,30,31,37)(20,34,33,35)(25,40,38,28)(26,41,32,42)(29,46,45,47)(39,43,48,44)Polycyclic presentation:
2^9|a^2=fi,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7268596);