Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3892 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)(5,6,9,14,10,13,20,27)(7,11,17,23,16,22,30,34)(8,12,19,26,18,25,31,36)(15,24,29,35,28,21,32,33);(2,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,24)(13,22)(14,23)(18,28)(19,29)(20,30)(21,25)(26,35)(27,34)(31,32)(33,36);(5,8)(6,12)(7,15)(9,19)(10,18)(11,21)(13,25)(14,26)(16,28)(17,29)(20,31)(22,24)(23,33)(27,36)(30,32)(34,35);(2,4)(8,18)(12,25)(15,28)(19,31)(21,24)(26,36)(29,32)(33,35)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72238736, #72239296, #72240678, … |
| maximal normal subgroup of | #72238736, #72239296, #72240678, … |
| maximal quotient of | #72238545, #72238754, #72279440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266704);