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 | #42674 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1390 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,29,34)(27,36,37,31)(28,33,40,48)(30,32,42,35)(38,49,50,44)(39,43,51,45)(41,46,53,47)(52,54,56,55);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,29,37)(26,31,34,36)(28,38,40,50)(30,39,42,51)(32,45,35,43)(33,44,48,49)(41,52,53,56)(46,55,47,54);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,32)(27,37)(28,40)(30,42)(31,43)(33,47)(34,35)(36,45)(44,55)(46,48)(49,54)(52,56);(1,3,4,7)(2,6,5,8)(11,18)(12,20)(14,22)(15,24)(25,28)(26,33)(27,38)(29,40)(30,41)(31,44)(32,46)(34,48)(35,47)(36,49)(37,50)(39,52)(42,53)(43,54)(45,55)(51,56);(2,5)(6,8)(11,18)(14,22)(17,19)(21,23)(26,34)(27,37)(28,40)(32,35)(39,51)(41,53)(44,49)(54,55)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=fi,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, 7288826);