Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,32)(29,42)(30,33)(31,39)(34,37)(36,47)(38,48)(40,46)(41,53)(43,50)(44,52)(45,51)(49,55)(54,56);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(27,36)(31,43)(35,47)(38,49)(39,50)(44,54)(48,55)(52,56);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,37,50,30,36,29,39)(26,31,42,47,33,43,34,35)(28,38,51,56,40,49,41,52)(32,44,53,55,46,54,45,48);(1,4,5,3)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,28,30,40)(26,32,33,46)(27,38,36,49)(29,41,37,51)(31,44,43,54)(34,45,42,53)(35,48,47,55)(39,52,50,56);(10,16)(11,18)(15,24)(17,19)(26,33)(27,36)(28,40)(34,42)(39,50)(41,51)(44,54)(48,55)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=fgi,d^2=fi,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=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6952365);