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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1543 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,41)(29,37)(31,36)(32,38)(33,39)(34,40)(35,43)(42,47)(44,46)(45,48);(2,6)(7,8)(10,13)(11,16)(17,19,21,30)(18,23,25,27)(20,28,32,43)(22,29,34,31)(24,35,38,41)(26,36,40,37)(33,44,45,42)(39,47,48,46);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20)(18,24)(19,28)(21,32)(22,33)(23,35)(25,38)(26,39)(27,41)(29,44)(30,43)(31,42)(34,45)(36,47)(37,46)(40,48);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(19,29)(20,32)(22,34)(23,36)(24,38)(26,40)(27,37)(28,42)(30,31)(35,46)(41,47)(43,44);(2,6)(7,8)(10,13)(14,15)(18,25)(20,32)(23,27)(26,40)(28,43)(33,45)(36,37)(42,44)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7456809);