Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324564 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26)(27,32)(28,38)(29,34)(30,45)(31,35)(33,39)(36,49)(37,46)(40,55)(41,47)(42,50)(43,56)(44,52)(48,54)(51,53);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,27,31,30)(26,32,35,45)(28,39,41,36)(29,42,44,37)(33,47,49,38)(34,50,52,46)(40,53,43,54)(48,55,51,56);(1,3)(2,5)(4,8)(6,7)(25,28,30,39,31,41,27,36)(26,33,32,38,35,49,45,47)(29,43,42,53,44,40,37,54)(34,51,46,56,52,48,50,55);(2,6)(3,8)(11,18)(14,17)(19,24)(22,23)(25,29)(26,34)(27,37)(28,40)(30,42)(31,44)(32,46)(33,48)(35,52)(36,53)(38,55)(39,54)(41,43)(45,50)(47,56)(49,51);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(29,44)(34,52)(37,42)(40,43)(46,50)(48,51)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7239480);