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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1382 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)(25,26)(27,33)(28,35)(29,45)(30,37)(31,36)(32,38)(34,44)(39,48)(40,54)(41,50)(42,49)(43,51)(46,52)(47,53)(55,56);(2,5)(3,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,27,32,43)(26,33,38,51)(28,39,29,40)(30,41,31,42)(34,47,52,56)(35,48,45,54)(36,49,37,50)(44,53,46,55);(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,28,32,29)(26,34,38,52)(27,39,43,40)(30,44,31,46)(33,47,51,56)(35,37,45,36)(41,53,42,55)(48,50,54,49);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(16,20)(19,24)(25,29,32,28)(26,35,38,45)(27,40,43,39)(30,44,31,46)(33,48,51,54)(34,37,52,36)(41,53,42,55)(47,50,56,49);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,30,32,31)(26,36,38,37)(27,41,43,42)(28,44,29,46)(33,49,51,50)(34,35,52,45)(39,53,40,55)(47,48,56,54)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,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, 7361723);