Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1271 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)(27,32)(28,39)(29,34)(30,45)(31,35)(33,43)(36,52)(37,47)(38,48)(40,54)(41,55)(42,51)(44,49)(46,53)(50,56);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,40)(29,44)(30,38)(31,37)(33,50)(34,49)(35,47)(36,41)(39,54)(42,53)(43,56)(45,48)(46,51)(52,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,29,42,31,41,30,43)(26,33,45,55,35,51,34,39)(27,36,44,56,37,40,38,53)(32,46,48,54,47,50,49,52);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(27,37)(28,41)(32,47)(33,51)(38,44)(39,55)(42,43)(48,49);(11,18)(14,22)(17,19)(21,23)(25,29,31,30)(26,34,35,45)(27,38,37,44)(28,42,41,43)(32,48,47,49)(33,39,51,55)(36,53,40,56)(46,54,50,52)Polycyclic presentation:
2^9|a^2=hi,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,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, 6418367);