Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1781 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62842 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3481 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)(5,17,19,9)(6,10)(7,13,21,31)(12,27)(14,26)(15,22)(16,24,18,23)(20,29,32,30)(25,28);(1,3)(2,7)(4,12)(5,13)(6,14)(8,20)(9,11)(10,21)(15,28)(16,29)(17,26)(18,30)(19,31)(22,32)(23,27)(24,25);(1,4)(2,8)(3,12)(5,18)(6,15)(7,20)(9,24)(10,22)(11,25)(13,30)(14,28)(16,19)(17,23)(21,32)(26,27)(29,31);(4,15)(8,22)(12,28)(16,18)(20,32)(23,24)(25,27)(29,30)Polycyclic presentation:
2^9|a^2=g,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fi,g^b=gh,d^c=di,e^c=ei,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72210464, #72210534, #72210561, … |
| maximal normal subgroup of | #72210464, #72210534, #72210561, … |
| maximal quotient of | #72232738, #72233183, #72233249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400587);