Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418473 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2673 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15263 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1708 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)(4,15)(5,19)(6,10)(7,13)(8,18)(9,21)(12,24)(14,26)(16,23)(17,25)(20,22)(27,28)(29,31)(30,32);(1,3)(2,7)(4,16)(5,20)(6,14)(8,24)(9,26)(10,22)(11,19)(12,17)(13,21)(15,28)(18,30)(23,25)(27,31)(29,32);(7,22)(11,26)(23,32)(24,28);(1,4,5,17,6,18,21,31)(2,8,9,25,10,15,19,29)(3,12,13,16,14,27,20,30)(7,23,11,24,22,32,26,28)Polycyclic presentation:
2^9|d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215995, #72216013, #72216142, … |
| maximal normal subgroup of | #72215995, #72216013, #72216142, … |
| maximal quotient of | C2×#418473, #72252110, #72265277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325669);