Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45091 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1889 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,29,37)(20,24,32,40)(21,25,34,42)(27,35,31,39)(28,36,30,38)(33,41,45,48)(43,46,44,47);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,23,26,37)(20,31,32,27)(21,28,34,30)(24,39,40,35)(25,36,42,38)(33,44,45,43)(41,47,48,46);(10,14)(11,16)(17,20)(18,24)(19,27)(21,33)(22,32)(23,35)(25,41)(26,40)(28,43)(29,31)(30,44)(34,45)(36,46)(37,39)(38,47)(42,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(19,28)(21,34)(23,36)(25,42)(27,43)(29,30)(31,44)(33,45)(35,46)(37,38)(39,47)(41,48);(2,5)(6,8)(10,14)(13,15)(19,29)(20,32)(23,37)(24,40)(28,30)(33,45)(36,38)(41,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7455808);