Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437473 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,17)(8,22)(9,14)(10,15)(11,16)(12,20)(13,23)(18,26)(19,28)(21,27)(24,29)(25,30)(31,32)(33,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,26)(14,17)(15,27)(16,28)(22,35)(23,32)(24,34)(25,33)(29,31)(30,36);(2,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,29)(15,22)(16,30)(17,31)(19,32)(20,33)(21,34)(27,35)(28,36);(2,4)(5,9,11,10)(6,14,16,15)(7,19,21,20)(8,23,24,25)(12,17,28,27)(13,29,30,22)(18,32,34,33)(26,31,36,35);(5,9,11,10)(6,15,16,14)(7,19,21,20)(8,23,24,25)(12,27,28,17)(13,22,30,29)(18,32,34,33)(26,35,36,31)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239541, #72251227, #72256257, … |
| maximal normal subgroup of | #72239541, #72251227, #72256257, … |
| maximal quotient of | #72239576, C2×#6437477, #72317023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344673);