Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| 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 | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,3,6,11,7,14)(4,9,16,12,17,25,22,24)(5,18,21,27,19,10,20,13)(15,23,29,26,28,30,31,32);(1,3,8,11,6,14,7,2)(4,9,16,12,17,25,22,24)(5,13,21,18,19,27,20,10)(15,23,29,26,28,30,31,32);(1,4)(2,9)(3,12)(5,15)(6,17)(7,22)(8,16)(10,23)(11,25)(13,26)(14,24)(18,30)(19,28)(20,31)(21,29)(27,32);(1,5)(2,10)(3,13)(4,15)(6,19)(7,20)(8,21)(9,23)(11,18)(12,26)(14,27)(16,29)(17,28)(22,31)(24,32)(25,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gi,e^c=ei,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72241171, #72241172, #72241303, … |
| maximal normal subgroup of | #72241171, #72241172, #72241303, … |
| maximal quotient of | C2×#364504 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271700);