Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368558 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14020 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1344 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)(3,11,14,7)(4,15)(5,18,19,9)(6,10)(8,17)(12,23,26,20)(13,21)(16,28,29,22)(24,27)(25,31)(30,32);(1,3)(2,7)(4,12,17,26)(5,13)(6,14)(8,20,15,23)(9,21)(10,11)(16,25,29,32)(18,24)(19,27)(22,30,28,31);(1,4)(2,8,10,15)(3,12,14,26)(5,16)(6,17)(7,20)(9,22,18,28)(11,23)(13,25,27,32)(19,29)(21,30)(24,31);(1,5)(2,9)(3,13)(4,16)(6,19)(7,21)(8,22)(10,18)(11,24)(12,25)(14,27)(15,28)(17,29)(20,30)(23,31)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=eg,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72237784, #72237793, #72237796, … |
| maximal normal subgroup of | #72237784, #72237793, #72237796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275754);