Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13584 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#353 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C4⋊Q8 | C24 |
| socle | C2 | C22×#353 |
| prime radical | #60639 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,11,21,28)(3,12,15,27,13,24,29,32)(4,8,17,25,18,9,20,26)(7,19,23,14,22,30,31,16);(1,3)(2,7)(4,14)(5,19)(6,13)(8,12)(9,24)(10,27)(11,22)(15,17)(16,18)(20,29)(21,30)(23,26)(25,31)(28,32);(3,13)(7,22)(12,24)(14,16)(15,29)(19,30)(23,31)(27,32);(1,4,6,18)(2,8,11,9)(3,14,13,16)(5,17,21,20)(7,12,22,24)(10,25,28,26)(15,30,29,19)(23,27,31,32)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ci,e^b=ehi,f^b=fgh,g^b=gi,f^e=fhi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72306620, #72306622, #72306625, … |
| maximal normal subgroup of | #72306620, #72306622, #72306625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419804);