Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405290 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14911 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2193 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(5,6)(10,22)(11,20)(13,21)(14,29)(15,16)(17,18)(23,32)(25,26)(27,30);(1,2)(3,13,16,21)(4,10)(5,20)(6,8)(7,11)(9,12,24,15)(14,28,30,31)(17,25)(18,22)(19,26)(23,27,32,29);(1,3)(2,9)(4,14)(5,12)(6,15)(7,16)(8,21)(10,23)(11,24)(13,20)(17,27)(18,29)(19,30)(22,31)(25,28)(26,32);(1,4)(2,10)(3,14)(5,17)(6,18)(7,19)(8,22)(9,23)(11,26)(12,27)(13,28)(15,29)(16,30)(20,25)(21,31)(24,32)Polycyclic presentation:
2^9|b^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72338515, #72338517, #72338524 |
| maximal normal subgroup of | #72338515, #72338517, #72338524 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312486);