Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419816 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15273 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,7)(4,16)(5,19)(6,9)(8,18)(10,20)(12,21,15,23)(13,24,27,22)(17,25)(26,31,28,32)(29,30);(1,3)(2,7)(4,13)(5,12)(6,14)(8,24)(9,11)(10,21)(15,20)(16,22)(17,26)(18,27)(19,23)(25,32)(28,30)(29,31);(3,12)(5,20)(7,21)(10,19)(11,23)(13,26)(14,15)(17,30)(22,31)(24,32)(25,29)(27,28);(1,4)(2,8,9,16)(3,13)(5,17)(6,18)(7,22,11,24)(10,25,19,29)(12,26)(14,27)(15,28)(20,30)(21,31,23,32)Polycyclic presentation:
2^9|a^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, #72240441, #72240445, … |
| maximal normal subgroup of | #72237642, #72240441, #72240445, … |
| maximal quotient of | #72237653, #72251195, #72255665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327012);