Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #364604 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13967 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2408 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)(3,12)(4,17,21,26)(5,14)(6,23)(7,10)(8,11)(9,20,27,25)(13,19,18,31)(15,22)(16,24)(28,29,32,30);(1,3)(2,5)(4,18,21,13)(6,15)(7,14)(8,16)(9,19,27,31)(10,24)(11,22)(12,23)(17,28,26,32)(20,29,25,30);(1,4)(2,9)(3,13)(5,19)(6,20)(7,25)(8,21)(10,26)(11,27)(12,28)(14,29)(15,30)(16,18)(17,23)(22,31)(24,32);(1,5)(2,3)(4,19)(6,24)(7,12)(8,22)(9,13)(10,15)(11,16)(14,23)(17,29)(18,27)(20,32)(21,31)(25,28)(26,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, #72240435, #72240440, … |
| maximal normal subgroup of | #72237642, #72240435, #72240440, … |
| maximal quotient of | #72237651, C2×#364604, #72251249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271800);