Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(6,11)(8,19)(9,23)(12,28)(14,24)(15,26)(16,32)(17,25)(20,27)(21,22)(29,31)(34,35);(5,6)(7,15)(8,20)(9,24)(10,14)(11,23)(12,18)(13,21)(16,33)(17,28)(19,30)(22,27)(25,26)(29,35)(31,32)(34,36);(5,7,9,25,10,18,23,17)(6,12,11,26,14,15,24,28)(8,16,19,32,22,34,21,35)(13,29,27,36,30,31,20,33);(5,8,10,22)(6,13,14,30)(7,16,18,34)(9,19,23,21)(11,27,24,20)(12,29,15,31)(17,35,25,32)(26,36,28,33)Polycyclic presentation:
2^9|a^2=h,c^2=eg,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265331, #72265644, #72265649 |
| maximal normal subgroup of | #72265331, #72265644, #72265649 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293113);