Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #435815 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15538 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1083 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)(3,11)(4,15,18,8)(5,19)(6,10)(7,13)(9,21)(12,24,27,28)(14,26)(16,22,30,32)(17,25,31,29)(20,23);(1,3)(2,7)(4,16)(5,20)(6,14)(8,24)(9,26)(10,23)(11,19)(12,17)(13,21)(15,28)(18,30)(22,25)(27,31)(29,32);(5,21)(9,19)(11,26)(13,20)(16,30)(17,31)(24,28)(25,29);(1,4)(2,8)(3,12,14,27)(5,17)(6,18)(7,22,23,32)(9,25)(10,15)(11,24,26,28)(13,16,20,30)(19,29)(21,31)Polycyclic presentation:
2^9|a^2=g,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285845, #72286246, #72286267 |
| maximal normal subgroup of | #72285845, #72286246, #72286267 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343011);