Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #386038 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14419 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60644 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1619 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)(3,14,17,10)(4,9)(5,8,16,24)(6,11)(7,12)(13,22)(15,25,31,29)(18,23,19,32)(20,26)(21,28)(27,30);(1,3,7,17)(2,8,12,24)(4,18,21,19)(5,22,16,6)(9,25,28,29)(10,13,14,11)(15,30,31,20)(23,26,32,27);(1,4,6,20,7,21,22,30)(2,9,13,27,12,28,11,26)(3,15,16,18,17,31,5,19)(8,23,10,25,24,32,14,29);(1,5,6,3,7,16,22,17)(2,10,11,8,12,14,13,24)(4,19,20,15,21,18,30,31)(9,25,26,23,28,29,27,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213409, #72216013, #72216511, … |
| maximal normal subgroup of | #72213409, #72216013, #72216511, … |
| maximal quotient of | C2×#386039, #72250477, #72255752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293235);