Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459299 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2752 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16668 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,20,28)(3,11,14,21,13,26,18,7)(4,8,16,24,17,25,19,27)(12,30,31,23,15,29,32,22);(7,21)(11,26)(22,23)(29,30);(1,3,5,18,6,13,20,14)(2,7,9,26,10,21,28,11)(4,15,16,31,17,12,19,32)(8,23,24,30,25,22,27,29);(1,4)(2,8,10,25)(3,12)(5,19)(6,17)(7,22,21,23)(9,27,28,24)(11,29,26,30)(13,15)(14,32)(16,20)(18,31)Polycyclic presentation:
2^9|a^2=f,c^2=ef,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338629, #72338632, #72338634, … |
| maximal normal subgroup of | #72338629, #72338632, #72338634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377106);