Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397242 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14747 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,8,3)(4,13,16,9,18,24,17,23)(5,14,19,10,20,26,21,25)(15,28,29,27,30,22,31,32);(1,3,6,11,7,12,8,2)(4,13,16,23,18,24,17,9)(5,14,19,25,20,26,21,10)(15,27,29,28,30,32,31,22);(1,4)(2,9,11,23)(3,13,12,24)(5,15)(6,16)(7,18)(8,17)(10,22,25,28)(14,27,26,32)(19,29)(20,30)(21,31);(1,5)(2,10)(3,14)(4,15)(6,19)(7,20)(8,21)(9,22)(11,25)(12,26)(13,27)(16,29)(17,31)(18,30)(23,28)(24,32)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215938, #72216141, #72216144, … |
| maximal normal subgroup of | #72215938, #72216141, #72216144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304438);