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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1231 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)(6,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,29,31)(27,35,37,48)(28,33,39,45)(30,42,41,34)(32,43,44,38)(36,49,50,56)(40,53,52,47)(46,54,55,51);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(28,39)(32,44)(34,42)(35,48)(36,50)(40,52)(51,54);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,37)(26,32,31,44)(28,36,39,50)(30,43,41,38)(33,46,45,55)(34,48,42,35)(40,54,52,51)(47,56,53,49);(1,3,4,8)(2,6,5,7)(12,20)(15,24)(17,19)(21,23)(25,28)(26,33)(27,36)(29,39)(30,40)(31,45)(32,46)(34,47)(35,49)(37,50)(38,51)(41,52)(42,53)(43,54)(44,55)(48,56);(10,16)(11,18)(15,24)(17,19)(28,39)(33,45)(36,50)(40,52)(46,55)(47,53)(49,56)(51,54)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=i,d^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=dh,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, 6942413);