Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,17)(11,19,21,15)(12,16,13,18)(20,24,22,23)(25,26)(27,37)(28,42)(29,35)(30,36)(31,47)(32,41)(33,43)(34,44)(38,54)(39,49)(40,51)(45,50)(46,53)(48,55)(52,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27)(26,32)(28,39)(29,45)(30,41)(31,40)(33,52)(34,49)(35,51)(36,37)(38,44)(42,54)(43,55)(46,56)(47,50)(48,53);(2,5)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,38)(29,43)(30,44)(31,46)(32,48)(34,35)(36,53)(37,52)(39,41)(40,55)(42,47)(45,56)(49,50)(51,54);(1,3,4,7)(2,6,5,8)(10,17)(15,19)(16,18)(23,24)(25,28)(26,34)(27,39)(29,46)(30,44)(31,43)(32,49)(33,35)(36,42)(37,54)(38,41)(40,55)(45,56)(47,53)(48,50)(51,52);(2,5)(6,8)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(25,29,30,31)(26,35,36,47)(27,40,41,45)(28,43,44,46)(32,50,37,51)(33,34,53,42)(38,55,39,56)(48,49,52,54)Polycyclic presentation:
2^9|a^2=fi,c^2=fi,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6684598);