Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453856 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16158 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,23,7,12,24,11)(3,13,17,27,18,26,16,8)(4,10,22,29,5,20,19,9)(14,28,15,30,21,25,32,31);(1,3,6,17,7,18,24,16)(2,8,23,13,12,27,11,26)(4,14,19,32,5,21,22,15)(9,28,20,31,29,25,10,30);(1,4,7,5)(2,9,12,29)(3,14)(6,22,24,19)(8,25)(10,11,20,23)(13,31)(15,17)(16,32)(18,21)(26,30)(27,28);(1,5,7,4)(2,10,12,20)(3,15)(6,22,24,19)(8,25)(9,11,29,23)(13,30)(14,17)(16,21)(18,32)(26,31)(27,28)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=ef,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251685, #72251692, #72251696, … |
| maximal normal subgroup of | #72251685, #72251692, #72251696, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361052);