Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324564 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,37)(28,34)(29,36)(30,32)(31,35)(33,41)(38,53)(39,46)(40,49)(42,47)(43,56)(44,52)(45,51)(48,54)(50,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(26,32)(28,42)(33,46)(35,36)(37,49)(38,43)(44,45)(54,55);(1,3)(2,5)(4,8)(6,7)(25,27,31,41,30,40,29,39)(26,33,36,37,32,46,35,49)(28,43,44,55,42,38,45,54)(34,50,51,56,47,48,52,53);(2,6)(3,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,34)(27,38)(29,45)(30,42)(31,44)(32,47)(33,48)(35,52)(36,51)(37,53)(39,55)(40,43)(41,54)(46,50)(49,56);(2,6)(5,7)(10,16)(14,17)(15,24)(22,23)(25,29,30,31)(26,35,32,36)(27,39,40,41)(28,44,42,45)(33,49,46,37)(34,51,47,52)(38,54,43,55)(48,53,50,56)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7236027);