Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1176 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,29,34,30,35,41,32)(27,31,37,44,38,45,51,42)(28,33,40,48,39,47,52,46)(36,43,50,55,49,54,56,53);(1,2)(3,7)(4,8)(5,6)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,38)(26,31,35,45)(28,36,39,49)(29,37,41,51)(32,42,34,44)(33,43,47,54)(40,50,52,56)(46,53,48,55);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(29,41)(31,42)(33,46)(34,35)(37,51)(40,52)(43,53)(44,45)(47,48)(50,56)(54,55);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28)(26,33)(27,36)(29,40)(30,39)(31,43)(32,46)(34,48)(35,47)(37,50)(38,49)(41,52)(42,53)(44,55)(45,54)(51,56);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(28,39)(33,47)(36,49)(40,52)(43,54)(46,48)(50,56)(53,55)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,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, 6782829);