Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1456 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)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,32)(21,25)(22,26)(23,30)(24,34)(28,37)(29,39)(31,38)(33,40)(35,45)(36,42)(41,46)(43,47)(44,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,33)(21,30)(22,29)(24,40)(25,27)(26,38)(28,34)(31,36)(32,37)(35,44)(39,42)(41,48)(43,46)(45,47);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20)(18,24)(19,28)(21,34)(22,35)(23,37)(25,32)(26,41)(27,40)(29,43)(30,33)(31,44)(36,46)(38,47)(39,48)(42,45);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(19,29)(22,36)(23,38)(26,42)(27,39)(28,43)(30,31)(33,44)(35,46)(37,47)(40,48)(41,45);(1,4,5,3)(2,7,8,6)(18,25)(19,30)(20,34)(26,42)(29,31)(35,46)(37,40)(47,48)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,d^2=fg,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7460956);