Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,33)(29,45)(30,31)(32,40)(34,37)(36,49)(38,53)(39,46)(41,47)(42,56)(43,52)(44,51)(48,54)(50,55);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(27,36)(28,41)(34,45)(39,40)(43,44)(48,50)(53,56);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,37,39,30,36,29,40)(26,32,45,49,31,46,34,35)(28,42,43,54,41,38,44,55)(33,50,51,53,47,48,52,56);(1,4,5,2)(3,6,8,7)(12,20)(15,24)(17,19)(21,23)(25,28)(26,33)(27,38)(29,44)(30,41)(31,47)(32,48)(34,52)(35,53)(36,42)(37,43)(39,55)(40,54)(45,51)(46,50)(49,56);(10,16)(11,18)(15,24)(17,19)(25,29,30,37)(26,34,31,45)(27,39,36,40)(28,43,41,44)(32,49,46,35)(33,51,47,52)(38,54,42,55)(48,53,50,56)Polycyclic presentation:
2^9|a^2=fh,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6951578);