Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453994 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16249 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2254 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)(3,14)(4,9)(5,12)(6,11)(7,13)(8,16)(10,18)(15,27)(17,25)(19,24)(20,31)(21,29)(22,30)(23,28)(26,32);(1,3)(2,8)(4,19)(5,18)(6,16)(7,17)(9,26)(10,13)(11,25)(12,14)(15,21)(20,22)(23,32)(24,29)(27,30)(28,31);(1,4)(2,9)(3,15)(5,23)(6,21)(7,22)(8,24)(10,31)(11,29)(12,28)(13,30)(14,26)(16,20)(17,32)(18,19)(25,27);(1,3)(2,10)(4,20)(5,18)(6,16)(7,17)(8,13)(9,27)(11,14)(12,25)(15,23)(19,22)(21,32)(24,28)(26,30)(29,31)Polycyclic presentation:
2^9|e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72215912, #72216059, #72216154, … |
| maximal normal subgroup of | #72215912, #72216059, #72216154, … |
| maximal quotient of | #72237194, C2×#453994, #72252209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361190);