Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16340 |
| 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 | |
1174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,13,7,12,6,11)(3,10,15,8,16,31,23,25)(4,17,20,28,22,9,21,29)(14,26,18,24,32,30,19,27);(1,3)(2,8,12,25)(4,18)(5,15)(6,23)(7,16)(9,26,17,30)(10,13,31,11)(14,21)(19,22)(20,32)(24,28,27,29);(1,4)(2,9)(3,14)(5,21)(6,20)(7,22)(8,24)(10,30)(11,29)(12,17)(13,28)(15,19)(16,32)(18,23)(25,27)(26,31);(1,3)(2,10,12,31)(4,19)(5,23)(6,15)(7,16)(8,13,25,11)(9,27,17,24)(14,21)(18,22)(20,32)(26,29,30,28)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=f,e^2=i,g^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,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251784, #72251791, #72251801, … |
| maximal normal subgroup of | #72251784, #72251791, #72251801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366644);