Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1256 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)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,29,34)(27,36,39,31)(28,33,41,51)(30,44,43,35)(32,45,49,42)(37,47,38,50)(40,48,56,53)(46,54,52,55);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,31)(28,37)(29,39)(30,40)(32,46)(33,50)(34,36)(35,48)(38,41)(42,54)(43,56)(44,53)(45,55)(47,51)(49,52);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,29,41)(26,32,34,49)(27,37,39,38)(30,45,43,42)(31,46,36,52)(33,35,51,44)(40,55,56,54)(47,53,50,48);(2,5)(6,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,33)(27,38)(29,41)(30,42)(31,47)(32,44)(34,51)(35,49)(36,50)(37,39)(40,55)(43,45)(46,48)(52,53)(54,56);(1,3,4,7)(2,6,5,8)(10,16)(11,18)(15,24)(17,19)(28,41)(32,49)(33,51)(37,38)(42,45)(46,52)(47,50)(54,55)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=i,e^2=fi,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=dhi,e^c=ei,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, 6945276);