Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394321 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2694 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)(7,17)(8,13)(9,14)(10,15)(11,16)(12,19)(18,29)(20,27)(21,28)(22,36)(23,30)(24,32)(25,31)(26,34)(33,35);(5,7,9,19,11,20,10,21)(6,12,15,17,16,28,14,27)(8,22,24,18,25,34,23,35)(13,29,30,36,31,33,32,26);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,26)(14,30)(15,32)(16,31)(17,33)(19,22)(20,35)(21,34)(27,29)(28,36);(2,4)(5,9,11,10)(6,14,16,15)(7,19,20,21)(8,23,25,24)(12,27,28,17)(13,30,31,32)(18,22,35,34)(26,29,36,33)Polycyclic presentation:
2^9|b^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239541, #72255613, #72256257, … |
| maximal normal subgroup of | #72239541, #72255613, #72256257, … |
| maximal quotient of | #72239575, C2×#394323, #72317027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301519);