Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #413935 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15159 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23407 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1411 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)(3,9)(4,17)(5,8)(6,11)(7,12)(10,18)(13,23)(14,28)(15,21)(16,24)(19,26)(20,25)(22,30)(27,32)(29,31);(2,8)(3,13)(5,6)(10,17)(11,12)(14,27)(15,16)(18,20)(21,23)(25,26)(28,31)(29,30);(1,3)(2,9)(4,14)(5,15)(6,13)(7,16)(8,21)(10,22)(11,23)(12,24)(17,31)(18,27)(19,29)(20,30)(25,32)(26,28);(1,4)(2,10,12,25)(3,14)(5,20)(6,18)(7,19)(8,17,11,26)(9,22,24,32)(13,27)(15,30)(16,29)(21,31,23,28)Polycyclic presentation:
2^9|d^2=gh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216013, #72216077, #72216203, … |
| maximal normal subgroup of | #72216013, #72216077, #72216203, … |
| maximal quotient of | C2×#413935, #72255670, #72255742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321131);