Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219498 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,10)(8,19,24,17)(9,25)(11,15)(12,16)(13,23,20,22)(14,26)(18,28)(21,33,36,29)(27,30)(31,34,32,35);(1,2,4,3)(5,7,11,16,12,15,10,6)(8,20,22,17,24,13,23,19)(9,18,27,30,28,25,26,14)(21,33,34,31,36,29,35,32);(5,8,12,24)(6,13)(7,17)(9,21,28,36)(10,22,11,23)(14,29)(15,19)(16,20)(18,31)(25,32)(26,34,27,35)(30,33);(1,3,4,2)(5,9,11,27,12,28,10,26)(6,14,7,18,16,30,15,25)(8,21,23,35,24,36,22,34)(13,29,17,31,20,33,19,32)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=fg,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72255889, #72255911, #72255919 |
| maximal normal subgroup of | #72255889, #72255911, #72255919 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276381);