Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474237 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16916 |
| commutator, Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
653 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)(4,15,17,8)(5,9,18,22)(6,10)(7,14)(12,19,26,23)(13,24,28,20)(16,29)(21,30)(25,31)(27,32);(1,3)(2,7,10,11)(4,12,17,26)(5,13)(6,14)(8,19)(9,20,22,24)(15,23)(16,25,30,27)(18,28)(21,31)(29,32);(1,4)(2,8)(3,12,14,26)(5,16)(6,17)(7,19,11,23)(9,21)(10,15)(13,27,28,25)(18,30)(20,32,24,31)(22,29);(1,5)(2,9,10,22)(3,13)(4,16)(6,18)(7,20,11,24)(8,21,15,29)(12,25)(14,28)(17,30)(19,31,23,32)(26,27)Polycyclic presentation:
2^9|a^2=eg,b^2=egh,c^2=e,d^2=f,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72318278, #72318288, #72318308, … |
| maximal normal subgroup of | #72318278, #72318288, #72318308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381433);