Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #372790 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14100 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1428 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)(3,8,14,25)(4,13)(5,10,20,28)(6,22)(7,11)(9,18)(12,19)(15,32,31,16)(17,29)(21,30,24,27)(23,26);(1,3)(2,8,11,25)(4,10,17,28)(5,13)(6,15)(7,14)(9,24)(12,16,22,32)(18,27,23,30)(19,31)(20,29)(21,26);(1,4)(2,9)(3,10)(5,16)(6,23)(7,17)(8,21)(11,26)(12,29)(13,22)(14,28)(15,30)(18,19)(20,32)(24,25)(27,31);(1,5,19,21,7,20,6,24)(2,10,22,27,11,28,12,30)(3,13,31,26,14,29,15,9)(4,16,18,8,17,32,23,25)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efg,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72219485, #72219486 |
| maximal normal subgroup of | #72213913, #72219485, #72219486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279986);