Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424600 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15323 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1242 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)(3,13,17,28)(4,9)(5,11)(6,10)(7,12)(8,16,23,15)(14,29,31,24)(18,22,30,32)(19,25)(20,26)(21,27);(1,3)(2,8)(4,18)(5,15)(6,16)(7,17)(9,24)(10,13)(11,28)(12,23)(14,20)(19,31)(21,30)(22,26)(25,32)(27,29);(1,4)(2,9)(3,14,17,31)(5,20)(6,19)(7,21)(8,22,23,32)(10,25)(11,26)(12,27)(13,24,28,29)(15,30,16,18);(2,10)(3,15)(5,6)(9,25)(11,12)(13,28)(14,30)(16,17)(18,31)(19,20)(24,29)(26,27)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72216071, #72216077, #72216217, … |
| maximal normal subgroup of | #72216071, #72216077, #72216217, … |
| maximal quotient of | C2×#424600, #72252172, #72252182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331796);