Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,16,24,15)(7,12,19,29,17,27,32,30)(8,20,22,13,23,11,21,26)(18,33,35,28,31,25,34,36);(6,11)(8,21)(9,24)(12,25)(13,14)(15,26)(16,20)(18,34)(19,32)(22,23)(27,33)(28,29)(30,36)(31,35);(2,4)(5,7)(6,12)(8,18)(9,19)(10,17)(11,25)(13,28)(14,29)(15,30)(16,27)(20,33)(21,34)(22,35)(23,31)(24,32)(26,36);(7,17)(12,27)(18,31)(19,32)(25,33)(28,36)(29,30)(34,35)Polycyclic presentation:
2^9|a^2=g,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ehi,g^b=gi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231480, #72231523, #72231551, … |
| maximal normal subgroup of | #72231480, #72231523, #72231551, … |
| maximal quotient of | #72257070, C2×#493528, #72310055, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400724);