Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12705 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1781 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C2 | C2×#1781 |
| prime radical | #62811 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3406 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)(4,8)(5,17,18,9)(6,10)(7,13,20,27)(12,24)(14,23)(15,28,29,21)(16,22)(19,25,30,32)(26,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,11)(10,20)(15,25)(16,26)(17,23)(18,27)(21,24)(22,30)(28,31)(29,32);(5,18)(9,17)(11,23)(13,27)(15,29)(21,28)(24,31)(25,32);(1,4,6,16)(2,8,10,22)(3,12,14,26)(5,15,18,29)(7,19,20,30)(9,21,17,28)(11,24,23,31)(13,25,27,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fi,g^b=gh,e^c=ei,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72218460, #72233438, #72233461, … |
| maximal normal subgroup of | #72218460, #72233438, #72233461, … |
| maximal quotient of | C2×#493390, C4×#12705, #72256657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400586);