Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #364625 |
| 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 | |
2387 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)(3,13)(4,17,21,9)(5,15)(6,12)(7,10)(8,11)(14,29,31,26)(16,24)(18,32,30,19)(20,27,25,28)(22,23);(1,3)(2,5)(4,18,21,30)(6,23)(7,15)(8,16)(9,26,17,29)(10,24)(11,22)(12,13)(14,20,31,25)(19,27,32,28);(1,4)(2,9)(3,14)(5,19)(6,20)(7,25)(8,21)(10,27)(11,17)(12,28)(13,29)(15,30)(16,31)(18,23)(22,32)(24,26);(1,5)(2,3)(4,19)(6,24)(7,13)(8,22)(9,14)(10,23)(11,16)(12,15)(17,31)(18,27)(20,26)(21,32)(25,29)(28,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215995, #72216130, #72216166, … |
| maximal normal subgroup of | #72215995, #72216130, #72216166, … |
| maximal quotient of | C2×#364625, #72255847, #72265243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271821);