Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,35)(27,32,39,46)(28,33,40,48)(29,43,30,34)(36,44,38,47)(37,45,52,56)(41,53,42,49)(50,54,51,55);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,32)(28,37)(29,38)(30,36)(31,39)(33,45)(34,44)(35,46)(40,52)(41,51)(42,50)(43,47)(48,56)(49,54)(53,55);(1,3,5,4)(2,6,7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,36)(28,40)(29,30)(32,44)(33,48)(34,43)(37,51)(38,39)(45,55)(46,47)(50,52)(54,56);(10,16)(14,17)(15,22)(21,23)(25,28,31,40)(26,33,35,48)(27,37,39,52)(29,41,30,42)(32,45,46,56)(34,49,43,53)(36,50,38,51)(44,54,47,55);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(21,23)(25,29)(26,34)(28,41)(30,31)(32,46)(33,49)(35,43)(36,38)(40,42)(45,56)(48,53)(50,51)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fg,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6629426);