Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396118 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
840 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,12,18,27)(8,13)(9,16)(10,14)(11,15)(17,33,34,25)(19,26,24,28)(20,29,35,36)(21,31)(22,30)(23,32);(1,2)(3,4)(5,7,10,19,11,18,9,24)(6,12,16,28,15,27,14,26)(8,20,23,34,21,35,22,17)(13,29,30,33,31,36,32,25);(2,4)(5,8)(6,13)(7,17,18,34)(9,22)(10,23)(11,21)(12,25,27,33)(14,32)(15,31)(16,30)(19,20,24,35)(26,36,28,29);(6,14,15,16)(8,21)(12,26,27,28)(13,30,31,32)(17,34)(20,35)(22,23)(25,29,33,36)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=f,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280622, #72280718 |
| maximal normal subgroup of | #72280622, #72280718 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 316144);