Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2712 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15404 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,8,5,7,12)(3,13,15,23,17,26,16,9)(4,18,21,20,22,28,24,10)(14,30,31,29,32,25,19,27);(1,3,6,16,8,17,7,15)(2,9,11,26,5,23,12,13)(4,19,21,32,22,31,24,14)(10,27,18,25,20,29,28,30);(1,4)(2,10)(3,14)(5,20)(6,24)(7,21)(8,22)(9,25)(11,28)(12,18)(13,29)(15,19)(16,31)(17,32)(23,30)(26,27);(1,5,8,2)(3,13)(4,20,22,10)(6,12,7,11)(9,16)(14,29)(15,23)(17,26)(18,21,28,24)(19,30)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338512, #72338514, #72338524 |
| maximal normal subgroup of | #72338512, #72338514, #72338524 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336800);