Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| 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 | #25502 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26)(27,34)(28,32)(29,38)(30,35)(31,36)(33,37)(39,40);(1,3,4,8)(2,5,6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27)(26,30)(28,35)(29,36)(31,33)(32,34)(37,39)(38,40);(3,8)(5,7)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(25,27)(26,31)(28,35)(29,39)(30,33)(32,40)(34,38)(36,37);(2,6)(5,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(26,32)(27,35)(33,38)(36,39);(11,19)(15,17)(18,24)(22,23)(27,35)(30,34)(31,40)(36,39)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fhi,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,g^c=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72282707, #72282801, #72283061, … |
| maximal normal subgroup of | #72282707, #72282801, #72283061, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6861603);