Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,32)(28,40)(29,34)(30,44)(31,35)(33,42)(36,45)(37,47)(38,49)(39,54)(41,56)(43,51)(46,52)(48,53)(50,55);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,31,38)(26,32,35,49)(28,41,29,37)(30,39,45,53)(33,50,51,46)(34,47,40,56)(36,48,44,54)(42,55,43,52);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28,31,29)(26,33,35,51)(27,37,38,41)(30,43,45,42)(32,46,49,50)(34,44,40,36)(39,55,53,52)(47,48,56,54);(3,8)(5,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,31,28)(26,34,35,40)(27,37,38,41)(30,43,45,42)(32,47,49,56)(33,44,51,36)(39,52,53,55)(46,48,50,54);(10,16)(14,17)(15,24)(22,23)(26,35)(27,38)(28,29)(36,44)(39,53)(42,43)(46,50)(47,56)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7032207);