Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14023 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60423 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1752 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)(4,8,17,24)(5,9)(6,11)(7,14)(10,19)(13,26,28,32)(15,25)(16,22,30,23)(18,21)(20,29,31,27);(1,3)(2,7)(4,13,17,28)(5,18)(6,15)(8,20,24,31)(9,12)(10,25)(11,21)(14,19)(16,29,30,27)(22,26,23,32);(1,4)(2,8,11,24)(3,13,15,28)(5,16)(6,17)(7,20)(9,22,10,23)(12,26)(14,27,18,29)(19,30)(21,31)(25,32);(2,9)(5,19)(7,12)(8,22)(10,11)(14,18)(16,30)(20,26)(21,25)(23,24)(27,29)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, C2×#368884, #72237690, … |
| maximal normal subgroup of | #72237642, C2×#368884, #72237690, … |
| maximal quotient of | C2×#368884, #72237655, #72237703 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276080);