Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2470 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)(7,12)(8,21,23,32)(9,15)(10,16)(11,17)(13,24,31,25)(14,20)(18,29,34,36)(19,28)(22,30,35,27)(26,33);(2,4)(5,7)(6,12)(8,22,23,35)(9,26)(10,19)(11,20)(13,29,31,36)(14,17)(15,33)(16,28)(18,25,34,24)(21,27,32,30);(5,8)(6,13)(7,18)(9,25)(10,24)(11,23)(12,27)(14,30)(15,32)(16,21)(17,31)(19,35)(20,34)(22,26)(28,36)(29,33);(2,4)(5,7)(6,14)(8,18)(9,26)(10,19)(11,20)(12,17)(13,30)(15,28)(16,33)(21,29)(22,25)(23,34)(24,35)(27,31)(32,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279945, #72285592, #72285620, … |
| maximal normal subgroup of | #72279945, #72285592, #72285620, … |
| maximal quotient of | #72302172, #72302173, #72303462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271719);