Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #413938 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15159 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1415 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)(14,23)(15,22)(17,24)(20,26)(21,25)(27,28)(29,31)(30,32);(1,3)(2,8)(4,15)(5,14)(6,16)(7,17)(9,22)(10,24)(11,13)(12,23)(18,27)(19,29)(20,30)(21,28)(25,32)(26,31);(3,14)(5,6)(8,13)(10,11)(15,28)(16,17)(18,26)(19,21)(22,27)(23,24)(29,30)(31,32);(1,4)(2,9,12,25)(3,15)(5,21)(6,19)(7,20)(8,22,23,32)(10,26,11,18)(13,27,24,31)(14,28)(16,29)(17,30)Polycyclic presentation:
2^9|d^2=gh,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,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216059, #72216077, #72216142, … |
| maximal normal subgroup of | #72216059, #72216077, #72216142, … |
| maximal quotient of | C2×#413938, #72252106 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321134);