Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 | #36019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1421 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,35)(27,31,37,44)(28,33,39,47)(29,32,41,34)(36,43,50,55)(38,42,51,45)(40,46,52,48)(49,53,56,54);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27)(26,31)(28,36)(29,38)(30,37)(32,42)(33,43)(34,45)(35,44)(39,50)(40,49)(41,51)(46,53)(47,55)(48,54)(52,56);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(29,41)(31,42)(33,46)(34,35)(38,51)(40,52)(43,53)(44,45)(47,48)(49,56)(54,55);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,39)(26,33,35,47)(27,36,37,50)(29,40,41,52)(31,43,44,55)(32,46,34,48)(38,49,51,56)(42,53,45,54);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(27,37)(28,39)(31,44)(33,47)(38,51)(40,52)(42,45)(46,48)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,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, 7110908);