Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3909 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)(3,11,14,7)(4,15,16,26)(5,10)(6,9)(8,17,24,18)(12,28)(13,27,19,22)(20,25)(21,30)(23,31)(29,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,23)(9,11)(10,22)(15,28)(16,29)(17,30)(18,31)(19,20)(21,24)(25,27)(26,32);(1,4)(2,8)(3,12)(5,18)(6,16)(7,21)(9,24)(10,26)(11,23)(13,31)(14,29)(15,25)(17,20)(19,30)(22,28)(27,32);(1,3)(2,7)(4,12)(5,19)(6,14)(8,21)(9,11)(10,27)(13,20)(15,28)(16,29)(17,31)(18,30)(22,25)(23,24)(26,32)Polycyclic presentation:
2^9|a^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72232886, #72234100, #72251416, … |
| maximal normal subgroup of | #72232886, #72234100, #72251416, … |
| maximal quotient of | #72232797, #72233314, #72234071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266730);