Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #451626 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16005 |
| 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 | |
804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,19,5,9,18,10)(3,11,27,29,13,7,14,22)(4,8,20,24,16,25,17,26)(12,28,15,23,30,21,31,32);(1,3,5,13)(2,7,9,11)(4,15)(6,14,18,27)(8,23)(10,22,19,29)(12,20)(16,31)(17,30)(21,26)(24,28)(25,32);(1,4,5,16)(2,8)(3,12,13,30)(6,20,18,17)(7,21)(9,25)(10,26)(11,28)(14,31,27,15)(19,24)(22,32)(23,29);(6,18)(7,11)(10,19)(14,27)(15,31)(17,20)(21,28)(24,26)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,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 | #72319162, #72319438, #72319453, … |
| maximal normal subgroup of | #72319162, #72319438, #72319453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358822);