Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1235 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)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26)(27,32)(28,38)(29,42)(30,44)(31,35)(33,40)(34,45)(36,50)(37,52)(39,53)(41,54)(43,47)(46,51)(48,55)(49,56);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27)(26,32)(28,39)(29,31)(30,37)(33,48)(34,47)(35,42)(36,41)(38,53)(40,55)(43,45)(44,52)(46,49)(50,54)(51,56);(11,17)(14,21)(18,23)(22,24)(25,28,30,40,31,41,45,56)(26,33,34,38,35,49,44,54)(27,36,37,51,29,39,43,55)(32,46,47,50,42,48,52,53);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(25,29,31,27)(26,32,35,42)(28,39,41,36)(30,43,45,37)(33,46,49,48)(34,47,44,52)(38,50,54,53)(40,55,56,51);(1,3,4,7)(2,6,5,8)(27,29)(32,42)(36,39)(37,43)(46,48)(47,52)(50,53)(51,55)Polycyclic presentation:
2^9|a^2=hi,c^2=g,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343644);