Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395191 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14885 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2629 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)(3,13)(4,18)(5,12)(6,10)(7,11)(8,15)(9,19)(14,23)(16,24)(17,22)(20,26)(21,25)(27,28)(29,31)(30,32);(1,3)(2,8)(4,14)(5,17)(6,15)(7,16)(9,23)(10,24)(11,22)(12,13)(18,27)(19,30)(20,29)(21,28)(25,31)(26,32);(8,22)(13,24)(23,31)(27,32);(1,4)(2,9,11,25)(3,14)(5,21)(6,19)(7,20)(8,23,22,31)(10,26,12,18)(13,27,24,32)(15,30)(16,29)(17,28)Polycyclic presentation:
2^9|d^2=g,e^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,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72215912, #72215972, … |
| maximal normal subgroup of | #72213514, #72215912, #72215972, … |
| maximal quotient of | #72234080, #72234083, #72239315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311208);