Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417617 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2678 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15948 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,20,5,8,6,11)(3,9,13,24,14,21,18,25)(4,15,19,10,16,26,17,22)(12,27,29,32,28,31,30,23);(2,8)(4,16)(11,20)(17,19);(1,3)(2,9)(4,12)(5,14)(6,18)(7,13)(8,21)(10,23)(11,25)(15,31)(16,28)(17,30)(19,29)(20,24)(22,32)(26,27);(1,4)(2,10)(3,12)(5,16)(6,19)(7,17)(8,22)(9,23)(11,26)(13,30)(14,28)(15,20)(18,29)(21,32)(24,31)(25,27)Polycyclic presentation:
2^9|a^2=eh,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216071, #72216130, … |
| maximal normal subgroup of | #72215958, #72216071, #72216130, … |
| maximal quotient of | #72237672, C2×#449600, #72252195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356796);