Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1778 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)(11,13)(12,14)(15,16)(17,18)(19,25)(20,34)(21,27)(22,28)(23,29)(24,30)(26,32)(31,40)(33,41)(35,47)(36,42)(37,43)(38,44)(39,45)(46,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,24)(21,36)(22,32)(23,33)(26,28)(27,42)(29,41)(30,34)(31,39)(35,38)(37,46)(40,45)(43,48)(44,47);(10,14)(11,15)(17,20,23,32)(18,26,29,34)(19,22,33,24)(21,35,38,36)(25,30,41,28)(27,42,44,47)(31,37,46,39)(40,45,48,43);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,24,39,23,38,22,37)(18,27,30,45,29,44,28,43)(19,31,32,36,33,46,20,35)(25,40,26,42,41,48,34,47);(2,5)(6,8)(17,22,23,24)(18,28,29,30)(19,32,33,20)(21,37,38,39)(25,26,41,34)(27,43,44,45)(31,36,46,35)(40,42,48,47)Polycyclic presentation:
2^9|c^2=i,d^2=fh,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7516718);