Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 | |
1341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26)(27,35)(28,33)(29,31)(30,43)(32,38)(34,42)(36,50)(37,49)(39,54)(40,45)(41,56)(44,51)(46,53)(47,55)(48,52);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(26,31)(27,36)(33,45)(34,43)(37,39)(38,51)(48,56)(53,55);(1,3)(2,5)(4,8)(6,7)(25,27,30,38,29,36,42,51)(26,32,34,35,31,44,43,50)(28,39,41,55,40,37,52,53)(33,47,48,54,45,46,56,49);(2,6)(3,8)(11,18)(14,17)(19,24)(22,23)(25,28,29,40)(26,33,31,45)(27,37,36,39)(30,41,42,52)(32,46,44,47)(34,48,43,56)(35,49,50,54)(38,53,51,55);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(26,31)(27,36)(28,40)(34,43)(38,51)(41,52)(46,47)(49,54)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7239555);