Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1438 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)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,37)(28,32)(29,34)(30,46)(31,36)(33,40)(35,38)(39,50)(41,47)(42,48)(43,55)(44,51)(45,56)(49,54)(52,53);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,38,42,31,40,30,28)(26,32,46,33,36,48,35,37)(29,39,53,43,44,54,45,41)(34,47,56,49,51,55,52,50);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,40,31,42,38,27)(26,33,35,32,36,37,46,48)(29,43,45,39,44,41,53,54)(34,50,52,55,51,49,56,47);(12,20)(15,24)(17,19)(21,23)(25,29)(26,34)(27,39)(28,41)(30,45)(31,44)(32,47)(33,49)(35,52)(36,51)(37,50)(38,53)(40,54)(42,43)(46,56)(48,55);(11,18)(14,22)(17,19)(21,23)(29,44)(34,51)(39,54)(41,43)(45,53)(47,55)(49,50)(52,56)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6417952);