Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15297 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,9,6,11)(3,13,17,24,16,28,14,8)(4,18,22,10,21,27,20,25)(15,23,19,29,31,32,30,26);(1,3,5,17,7,16,6,14)(2,8,12,13,9,24,11,28)(4,19,22,31,21,30,20,15)(10,26,27,23,25,29,18,32);(2,9)(3,14)(5,6)(8,13)(10,25)(15,30)(16,17)(19,31)(20,22)(23,29)(24,28)(26,32);(1,4)(2,10)(3,15)(5,22)(6,20)(7,21)(8,23)(9,25)(11,18)(12,27)(13,29)(14,30)(16,31)(17,19)(24,32)(26,28)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251787, #72251789, #72251801, … |
| maximal normal subgroup of | #72251787, #72251789, #72251801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328960);