Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454389 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2815 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16660 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1152 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)(4,17)(5,12)(6,10)(7,11)(8,16)(9,19)(14,25)(15,28)(18,24)(20,27)(21,26)(22,23)(29,31)(30,32);(8,23)(13,28)(24,25)(29,32);(1,3,5,16,7,15,6,22)(2,8,10,13,11,23,12,28)(4,18,19,31,20,14,21,30)(9,25,17,29,26,24,27,32);(1,4)(2,9,11,26)(3,14)(5,21)(6,19)(7,20)(8,24,23,25)(10,27,12,17)(13,29,28,32)(15,18)(16,31)(22,30)Polycyclic presentation:
2^9|c^2=f,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237054, #72237088, #72237090, … |
| maximal normal subgroup of | #72237054, #72237088, #72237090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376878);