Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394271 |
| 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 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2681 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)(7,11)(8,18)(9,22)(10,14)(12,21)(13,24)(15,30)(16,32)(17,27)(19,35)(20,29)(23,28)(25,31)(26,34)(33,36);(5,7,10,17)(6,11,14,27)(8,19)(9,16,24,34)(12,25)(13,26,22,32)(15,20)(18,30)(21,33)(23,31)(28,36)(29,35);(5,8)(6,12)(7,15)(9,23)(10,20)(11,25)(13,29)(14,28)(16,33)(17,19)(18,22)(21,24)(26,35)(27,36)(30,32)(31,34);(2,4)Polycyclic presentation:
2^9|b^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72316838, #72316858 |
| maximal normal subgroup of | #72316838, #72316858 |
| maximal quotient of | #72316883, #72317003, #72317028 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301468);