Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682380 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26)(27,31)(28,40)(29,33)(30,35)(32,43)(34,42)(36,52)(37,46)(38,47)(39,51)(41,55)(44,49)(45,54)(48,53)(50,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27)(26,31)(28,41)(29,39)(30,37)(32,48)(33,51)(34,47)(35,46)(36,44)(38,42)(40,55)(43,53)(45,50)(49,52)(54,56);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,42,56,30,44,29,43)(26,32,33,49,35,50,34,40)(27,36,38,53,37,41,39,54)(31,45,51,55,46,48,47,52);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(27,37)(31,46)(36,41)(38,39)(45,48)(47,51)(52,55)(53,54);(3,7)(5,8)(11,21)(15,18)(19,20)(23,24)(25,29,30,42)(26,33,35,34)(27,38,37,39)(28,43,44,56)(31,47,46,51)(32,49,50,40)(36,53,41,54)(45,52,48,55)Polycyclic presentation:
2^9|a^2=f,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670881);