Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #398104 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14759 |
| 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 | |
1364 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)(3,7)(4,8,16,24)(5,9)(6,10)(11,18)(12,19,15,28)(13,21)(14,20)(17,23,30,25)(22,27)(26,31,29,32);(1,3)(2,7)(4,15,16,12)(5,14)(6,13)(8,19,24,28)(9,20)(10,21)(11,22)(17,29,30,26)(18,27)(23,31,25,32);(1,4)(2,8,10,24)(3,12)(5,17)(6,16)(7,19,21,28)(9,23,11,25)(13,15)(14,26)(18,30)(20,31,22,32)(27,29);(2,9)(5,18)(7,20)(8,23)(10,11)(14,27)(17,30)(19,31)(21,22)(24,25)(26,29)(28,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, C2×#398104, #72237692, … |
| maximal normal subgroup of | #72237642, C2×#398104, #72237692, … |
| maximal quotient of | C2×#398104, #72237707 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305300);