Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,32,38)(27,34,40,49)(28,33,41,48)(29,35,44,52)(30,37,31,36)(39,50,55,51)(42,47,43,56)(45,54,46,53);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,31,41,32,40,30,28)(26,33,37,49,38,48,36,34)(29,39,46,43,44,55,45,42)(35,47,54,51,52,56,53,50);(11,17)(14,21)(18,19)(22,23)(25,28,30,40,32,41,31,27)(26,34,36,48,38,49,37,33)(29,43,45,39,44,42,46,55)(35,51,53,47,52,50,54,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29)(26,35)(27,39)(28,42)(30,45)(31,46)(32,44)(33,47)(34,50)(36,53)(37,54)(38,52)(40,55)(41,43)(48,56)(49,51);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(27,40)(28,41)(29,44)(33,48)(34,49)(35,52)(45,46)(53,54)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=h,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,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, 6394613);