Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
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)(27,36)(28,41)(29,33)(30,35)(31,40)(32,42)(34,37)(38,52)(39,46)(43,47)(44,53)(45,50)(48,51)(49,54)(55,56);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,37,50,30,39,29,40)(26,31,33,46,35,45,34,36)(28,38,51,56,43,53,42,54)(32,44,47,55,48,52,41,49);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,29,42,30,43,37,51)(26,32,34,41,35,48,33,47)(27,38,40,54,39,53,50,56)(31,44,36,49,45,52,46,55);(12,20)(15,24)(17,19)(21,23)(27,39)(31,45)(36,46)(38,53)(40,50)(44,52)(49,55)(54,56);(11,18)(14,22)(17,19)(21,23)(25,29,30,37)(26,33,35,34)(27,40,39,50)(28,42,43,51)(31,46,45,36)(32,47,48,41)(38,54,53,56)(44,55,52,49)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6417913);