Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,34)(27,31,38,45)(28,32,39,46)(29,33,41,48)(35,42,37,44)(36,43,51,55)(40,47,52,56)(49,53,50,54);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,30,38)(26,31,34,45)(28,36,39,51)(29,35,41,37)(32,43,46,55)(33,42,48,44)(40,49,52,50)(47,53,56,54);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(27,35)(29,41)(31,42)(33,48)(36,49)(37,38)(40,52)(43,53)(44,45)(47,56)(50,51)(54,55);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28)(26,32)(27,36)(29,40)(30,39)(31,43)(33,47)(34,46)(35,49)(37,50)(38,51)(41,52)(42,53)(44,54)(45,55)(48,56);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,35)(28,39)(29,41)(31,42)(32,46)(33,48)(36,50)(37,38)(43,54)(44,45)(49,51)(53,55)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fg,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6597240);