Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,25)(17,27)(22,36)(23,32)(24,31)(28,34)(29,33)(30,37)(35,39)(38,40);(2,4)(5,6)(7,8)(9,11)(10,16)(12,22)(13,24)(14,25)(15,23)(17,30)(18,21)(19,32)(20,31)(26,40)(27,38)(28,39)(29,37)(33,35)(34,36);(6,8)(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,30)(18,33)(19,26)(20,34)(21,35)(23,37)(24,38)(25,39)(31,36)(32,40);(11,23)(16,31)(21,32)(22,37)(24,25)(30,36)(35,40)(38,39);(9,13,15,14)(10,18,20,19)(11,24,23,25)(12,27,29,28)(16,21,31,32)(17,33,34,26)(22,38,37,39)(30,35,36,40)Polycyclic presentation:
2^9|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,f^b=fi,g^b=gi,f^c=fi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314510, #72314518, #72314530, … |
| maximal normal subgroup of | #72314510, #72314518, #72314530, … |
| maximal quotient of | #72321559 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338959);