Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#102 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1456 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)(5,6,10,15,24,28,26,14,11,16,23,34,8,13,9,25)(7,17,20,12,18,30,19,29,22,35,36,32,21,33,27,31);(5,7,8,18,11,22,24,21)(6,12,28,31,16,32,13,29)(9,20,23,19,26,36,10,27)(14,33,34,35,25,30,15,17);(2,4);(5,8,11,24)(6,13,16,28)(7,18,22,21)(9,23,26,10)(12,29,32,31)(14,15,25,34)(17,30,35,33)(19,36,27,20)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,g^b=gh,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249892, #72249903, #72249913, … |
| maximal normal subgroup of | #72249892, #72249903, #72249913, … |
| maximal quotient of | #72265731, #72285512, #72285975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401668);