Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455397 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2700 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16263 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,10)(3,13,15,25,16,8,22,24)(4,17,19,9,21,27,20,28)(14,26,30,29,31,32,18,23);(1,3)(2,8)(4,18)(5,22)(6,15)(7,16)(9,26)(10,24)(11,13)(12,25)(14,20)(17,29)(19,31)(21,30)(23,27)(28,32);(1,4)(2,9,11,28)(3,14)(5,19)(6,20)(7,21)(8,23,13,29)(10,17,12,27)(15,30)(16,31)(18,22)(24,26,25,32);(2,10,11,12)(8,24,13,25)(9,27,28,17)(23,32,29,26)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319923, #72319935, #72320044, … |
| maximal normal subgroup of | #72319923, #72319935, #72320044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362593);