Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | #2011 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1407 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,31)(28,33)(29,32)(30,35)(34,42)(36,45)(37,44)(38,46)(39,48)(40,50)(41,47)(43,51)(49,54)(52,56)(53,55);(2,6)(3,4)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,36)(26,31,35,45)(28,39,40,37)(29,38,42,51)(32,46,34,43)(33,48,50,44)(41,53,54,52)(47,55,49,56);(1,2)(3,8)(4,7)(5,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(27,36)(28,40)(29,42)(31,43)(33,49)(34,35)(44,56)(45,46)(47,50)(48,55)(52,53);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,40)(26,33,35,50)(27,37,36,39)(29,41,42,54)(31,44,45,48)(32,47,34,49)(38,52,51,53)(43,55,46,56);(1,4,5,3)(2,7,6,8)(10,18)(15,19)(16,17)(23,24)(27,36)(28,40)(31,45)(33,50)(38,51)(41,54)(43,46)(47,49)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7396492);