Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31282 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2684 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14357 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1927 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)(3,15,18,28)(4,19,22,10)(5,23)(6,12)(7,14)(8,13)(9,16,27,17)(11,20)(21,32,26,30)(24,29)(25,31);(1,3)(2,9)(4,20)(5,21)(6,17)(7,16)(8,18)(10,29)(11,30)(12,28)(13,27)(14,15)(19,23)(22,25)(24,26)(31,32);(1,4)(2,10)(3,5)(6,21)(7,26)(8,22)(9,11)(12,30)(13,19)(14,32)(15,29)(16,20)(17,25)(18,24)(23,28)(27,31);(1,5)(2,11)(3,4)(6,25)(7,20)(8,24)(9,10)(12,23)(13,31)(14,29)(15,32)(16,26)(17,21)(18,22)(19,27)(28,30)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72215981, #72219482, #72219488, … |
| maximal normal subgroup of | #72215981, #72219482, #72219488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290253);