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 | |
1363 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,30,35)(27,31,38,34)(28,39,41,43)(29,33,42,48)(32,45,47,36)(37,44,51,49)(40,52,53,54)(46,55,56,50);(1,2)(3,6)(4,5)(7,8)(11,18)(12,21)(14,17)(15,20)(25,27,30,38)(26,31,35,34)(28,36,41,45)(29,37,42,51)(32,43,47,39)(33,44,48,49)(40,50,53,55)(46,54,56,52);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,28,38,45,30,41,27,36)(26,32,31,43,35,47,34,39)(29,40,51,55,42,53,37,50)(33,46,44,54,48,56,49,52);(2,5)(6,8)(11,18)(14,17)(19,24)(22,23)(25,29)(26,33)(27,37)(28,40)(30,42)(31,44)(32,46)(34,49)(35,48)(36,50)(38,51)(39,52)(41,53)(43,54)(45,55)(47,56);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(28,41)(29,42)(32,47)(33,48)(36,45)(37,51)(39,43)(44,49)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=fghi,e^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=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, 7103531);