Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470149 |
| 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 | |
1352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,36)(28,41)(29,34)(30,35)(31,44)(32,39)(33,42)(37,52)(38,51)(40,47)(43,48)(45,49)(46,55)(50,53)(54,56);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,37)(29,38)(30,39)(31,40)(33,46)(34,47)(35,36)(41,56)(42,53)(43,54)(44,51)(45,55)(48,52)(49,50);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,30,43)(26,33,35,49)(27,37,39,54)(29,42,31,45)(32,46,36,50)(34,48,44,41)(38,53,40,55)(47,52,51,56);(11,18)(14,22)(17,19)(21,23)(25,29,30,31)(26,34,35,44)(27,38,39,40)(28,42,43,45)(32,47,36,51)(33,48,49,41)(37,53,54,55)(46,52,50,56);(1,3,4,7)(2,6,5,8)(10,16)(14,22)(15,24)(21,23)(27,39)(28,43)(32,36)(33,49)(38,40)(41,48)(42,45)(47,51)Polycyclic presentation:
2^9|a^2=fi,c^2=hi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6386422);