Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363956 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13956 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62855 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2621 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)(3,7,12,22)(4,14,16,27)(5,10)(6,9)(8,15,24,18)(11,28)(13,21,17,26)(19,25)(20,31)(23,30)(29,32);(1,3)(2,7,9,22)(4,11)(5,17)(6,12)(8,23,24,20)(10,26,25,21)(13,19)(14,28,27,32)(15,31)(16,29)(18,30);(1,4)(2,8)(3,11)(5,18)(6,16)(7,20)(9,24)(10,27)(12,29)(13,31)(14,25)(15,19)(17,30)(21,32)(22,23)(26,28);(1,3)(2,7,9,22)(4,11)(5,13)(6,12)(8,20,24,23)(10,21,25,26)(14,28,27,32)(15,30)(16,29)(17,19)(18,31)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72234100, #72251419, #72251422, … |
| maximal normal subgroup of | #72234100, #72251419, #72251422, … |
| maximal quotient of | #72234064, #72234077, #72256102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271152);