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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,32,37)(27,33,40,51)(28,41,29,35)(30,36,31,46)(34,44,53,43)(38,48,42,56)(39,49,45,50)(47,54,52,55);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,33)(28,42)(29,38)(30,45)(31,39)(32,40)(34,52)(35,48)(36,50)(37,51)(41,56)(43,54)(44,55)(46,49)(47,53);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28)(26,34)(27,38)(29,32)(30,43)(31,44)(33,47)(35,36)(37,53)(39,54)(40,42)(41,46)(45,55)(48,49)(50,56)(51,52);(3,8)(5,7)(11,18)(12,20)(14,22)(15,24)(25,29,32,28)(26,35,37,41)(27,38,40,42)(30,43,31,44)(33,48,51,56)(34,46,53,36)(39,55,45,54)(47,50,52,49);(2,6)(5,7)(11,18)(14,22)(17,19)(21,23)(25,30,32,31)(26,36,37,46)(27,39,40,45)(28,43,29,44)(33,49,51,50)(34,35,53,41)(38,54,42,55)(47,48,52,56)Polycyclic presentation:
2^9|a^2=fh,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,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, 7247098);