Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427841 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2778 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15958 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,12)(3,13,15,25,16,8,22,24)(4,17,19,9,20,27,21,28)(14,29,31,23,18,32,30,26);(1,3,5,15,7,16,6,22)(2,8,12,24,11,13,10,25)(4,18,19,31,20,14,21,30)(9,26,27,32,28,23,17,29);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,23)(10,27)(11,28)(12,17)(13,26)(15,31)(16,18)(22,30)(24,32)(25,29);(2,10,11,12)(8,24,13,25)(9,27,28,17)(23,32,26,29)Polycyclic presentation:
2^9|a^2=eh,b^2=ef,d^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,d^b=de,g^b=gi,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318430, #72318876, #72318991, … |
| maximal normal subgroup of | #72318430, #72318876, #72318991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357325);