Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #376205 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14175 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1576 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)(4,16,19,9)(5,15)(6,20)(7,12)(8,10)(11,22)(14,17,32,30)(18,27,21,29)(23,26,25,31)(24,28);(1,3,6,10,7,15,22,28)(2,8,11,13,12,24,20,5)(4,14,18,31,19,32,21,26)(9,25,27,30,16,23,29,17);(1,4)(2,9)(3,14)(5,17)(6,21)(7,19)(8,23)(10,26)(11,29)(12,16)(13,30)(15,32)(18,22)(20,27)(24,25)(28,31);(1,5,6,8,7,13,22,24)(2,10,11,15,12,28,20,3)(4,17,21,23,19,30,18,25)(9,26,29,32,16,31,27,14)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=eh,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, #72240441, #72240442, … |
| maximal normal subgroup of | #72237642, #72240441, #72240442, … |
| maximal quotient of | #72237654, #72251193, C2×#376205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283401);