Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2709 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15561 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1180 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)(3,7,13,11)(4,15,16,25)(5,10)(6,9)(8,17,23,18)(12,27)(14,21,30,26)(19,24)(20,31)(22,29)(28,32);(1,3)(2,7,9,11)(4,12,16,28)(5,14)(6,13)(8,22)(10,21,24,26)(15,32)(17,29,18,31)(19,30)(20,23)(25,27);(1,2)(3,11)(4,8)(5,10)(6,9)(7,13)(12,22)(14,26)(15,17)(16,23)(18,25)(19,24)(20,28)(21,30)(27,31)(29,32);(1,4)(2,8)(3,12,13,28)(5,18)(6,16)(7,20,11,22)(9,23)(10,25)(14,31,30,29)(15,24)(17,19)(21,32,26,27)Polycyclic presentation:
2^9|a^2=g,b^2=ef,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72283686, #72319988, #72320009, … |
| maximal normal subgroup of | #72283686, #72319988, #72320009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343669);