Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403153 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14868 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,5,10,6,19)(3,8,14,21,15,23,12,24)(4,16,20,30,17,9,18,25)(13,27,28,32,29,22,26,31);(1,3)(2,8,10,23)(4,13)(5,15)(6,12)(7,14)(9,22,16,27)(11,21,19,24)(17,29)(18,26)(20,28)(25,31,30,32);(3,12)(6,7)(8,21)(11,19)(13,26)(14,15)(18,20)(22,31)(23,24)(25,30)(27,32)(28,29);(1,4)(2,9,10,16)(3,13)(5,17)(6,18)(7,20)(8,22,23,27)(11,25,19,30)(12,26)(14,28)(15,29)(21,31,24,32)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72319853 |
| maximal normal subgroup of | #72319853 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310349);