Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,17,18,14)(12,20,22,15)(19,23,21,24)(25,26)(27,37)(28,34)(29,36)(30,32)(31,35)(33,41)(38,49)(39,53)(40,46)(42,47)(43,56)(44,52)(45,51)(48,54)(50,55);(1,3)(2,6)(4,7)(5,8)(9,11,13,18)(10,14,16,17)(12,21,22,19)(15,24,20,23)(26,32)(27,38)(28,42)(35,36)(40,41)(44,45)(48,50)(53,56);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,17)(20,24)(21,22)(25,27,31,41,30,38,29,40)(26,33,36,37,32,46,35,49)(28,43,44,55,42,39,45,54)(34,50,51,56,47,48,52,53);(3,8)(6,7)(11,18)(14,17)(19,21)(23,24)(25,28)(26,34)(27,39)(29,45)(30,42)(31,44)(32,47)(33,48)(35,52)(36,51)(37,53)(38,43)(40,55)(41,54)(46,50)(49,56);(9,12,13,22)(10,15,16,20)(11,19,18,21)(14,23,17,24)(25,29,30,31)(26,35,32,36)(27,40,38,41)(28,44,42,45)(33,49,46,37)(34,51,47,52)(39,54,43,55)(48,53,50,56)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=fgi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093404);