Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6374187 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2308 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11,14,7)(4,15,17,8)(5,9)(6,10)(12,19)(13,24,27,20)(16,28,29,21)(18,22)(23,26)(25,30)(31,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,20)(10,11)(15,23)(16,25)(17,26)(18,27)(21,30)(22,24)(28,31)(29,32);(5,18)(7,11)(9,22)(13,27)(16,29)(19,23)(21,28)(25,32);(1,4)(2,8)(3,12)(5,16)(6,17)(7,19)(9,21)(10,15)(11,23)(13,25)(14,26)(18,29)(20,30)(22,28)(24,31)(27,32);(1,5,6,18)(2,9,10,22)(3,13,14,27)(4,16,17,29)(7,20,11,24)(8,21,15,28)(12,25,26,32)(19,30,23,31)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,c^b=cf,g^b=gi,h^b=hi,e^c=ei,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72266804, #72266806, #72266931, … |
| maximal normal subgroup of | #72266804, #72266806, #72266931, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281384);