Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391213 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14621 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1128 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)(3,12,15,7)(4,9)(5,8)(6,10)(11,20)(13,29,30,23)(14,24,19,22)(16,32,31,26)(17,27)(18,28)(21,25);(1,3)(2,7)(4,16)(5,19)(6,15)(8,24)(9,26)(10,12)(11,22)(13,17)(14,20)(18,31)(21,30)(23,27)(25,29)(28,32);(2,8)(5,20)(7,22)(9,27)(10,11)(12,24)(14,19)(16,31)(17,21)(23,26)(25,28)(29,32);(1,4)(2,9)(3,13)(5,21)(6,18)(7,23)(8,25)(10,28)(11,27)(12,29)(14,31)(15,30)(16,19)(17,20)(22,32)(24,26)Polycyclic presentation:
2^9|a^2=h,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,f^c=fi,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237063, #72237071, #72237092, … |
| maximal normal subgroup of | #72237063, #72237071, #72237092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298409);