Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #451759 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16012 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
941 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,12)(4,16)(5,7,20,10)(8,22)(9,17)(13,24)(14,23,28,32)(15,25)(18,26)(19,21)(27,29,31,30);(2,7)(3,13)(4,17)(5,20)(8,12)(10,11)(14,29)(16,21)(18,19)(24,25)(27,31)(28,30);(1,3,5,15,6,13,20,22)(2,8,10,24,11,25,7,12)(4,14,17,30,19,28,18,29)(9,23,16,31,26,32,21,27);(1,4,5,17,6,19,20,18)(2,9,7,21,11,26,10,16)(3,14,15,30,13,28,22,29)(8,23,12,27,25,32,24,31)Polycyclic presentation:
2^9|a^2=egh,c^2=e,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251685, #72251691, #72251700, … |
| maximal normal subgroup of | #72251685, #72251691, #72251700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358955);