Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,17)(7,18,13,23)(8,21,24,20)(9,25)(10,16,11,15)(14,28)(19,35)(22,30)(26,31)(27,33)(29,32)(34,36);(1,2)(3,4)(5,7,10,20,12,13,11,21)(6,8,15,18,17,24,16,23)(9,26,27,30,29,36,28,19)(14,31,32,35,33,34,25,22);(2,4)(5,8)(6,13)(7,17)(9,22)(10,23)(11,18)(12,24)(14,30)(15,20)(16,21)(19,33)(25,36)(26,32)(27,34)(28,31)(29,35);(5,9)(6,14)(7,19)(8,22)(10,28)(11,27)(12,29)(13,30)(15,25)(16,32)(17,33)(18,34)(20,36)(21,26)(23,31)(24,35)Polycyclic presentation:
2^9|a^2=ef,b^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72323003 |
| maximal normal subgroup of | #72323003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326403);