Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460711 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16370 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
953 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)(3,12,15,23)(4,16)(5,9)(6,11)(7,14,24,31)(8,18)(10,21)(13,25,17,22)(19,27)(20,26)(28,30,32,29);(1,3,6,15)(2,7)(4,17)(5,14,21,31)(8,25,27,22)(9,12)(10,23)(11,24)(13,20)(16,32,26,28)(18,30)(19,29);(1,4,5,19,6,20,21,18)(2,8,9,16,11,27,10,26)(3,13,14,30,15,17,31,29)(7,22,12,28,24,25,23,32);(4,18)(5,21)(8,26)(9,10)(12,23)(13,29)(14,31)(16,27)(17,30)(19,20)(22,32)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251685, #72251693, #72251697, … |
| maximal normal subgroup of | #72251685, #72251693, #72251697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367907);