Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15257 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2399 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)(5,11)(6,10)(7,12)(8,16)(9,19)(14,25)(15,23)(17,24)(20,27)(21,26)(22,31)(28,30)(29,32);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,25)(10,24)(11,13)(12,23)(18,28)(19,29)(20,31)(21,30)(22,26)(27,32);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,18)(11,27)(12,26)(13,28)(15,29)(16,30)(17,31)(23,25)(24,32);(2,10)(3,15)(5,6)(8,23)(9,18)(11,12)(14,29)(16,17)(19,21)(22,25)(26,27)(30,31)Polycyclic presentation:
2^9|e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72216037, #72216059, #72216112, … |
| maximal normal subgroup of | #72216037, #72216059, #72216112, … |
| maximal quotient of | #72237171, #72237509, #72237511, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324932);