Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13967 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,19)(3,12,14,8,15,24,20,22)(4,16,17,9,18,26,21,25)(13,23,30,32,29,31,28,27);(1,3,6,14,7,15,5,20)(2,8,10,24,11,22,19,12)(4,13,17,30,18,29,21,28)(9,23,26,32,25,31,16,27);(8,22)(12,24)(23,31)(27,32);(1,4)(2,9)(3,13)(5,21)(6,17)(7,18)(8,23)(10,26)(11,25)(12,27)(14,30)(15,29)(16,19)(20,28)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,h^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,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213409, #72215936, #72215958, … |
| maximal normal subgroup of | #72213409, #72215936, #72215958, … |
| maximal quotient of | C2×#364596, #72250500, #72255966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271792);