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 | #62842 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3917 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,10,5,11)(3,13,15,27,16,8,21,23)(4,17,20,9,18,26,19,25)(14,28,30,22,29,32,31,24);(1,3)(2,8)(4,14)(5,15)(6,21)(7,16)(9,24)(10,13)(11,23)(12,27)(17,28)(18,29)(19,30)(20,31)(22,25)(26,32);(1,4)(2,9)(3,14)(5,20)(6,19)(7,18)(8,22)(10,25)(11,26)(12,17)(13,24)(15,31)(16,29)(21,30)(23,28)(27,32);(2,10)(5,6)(8,13)(9,25)(15,21)(19,20)(22,24)(30,31)Polycyclic presentation:
2^9|a^2=h,f^2=i,h^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 | #72235139, #72238713, #72238720, … |
| maximal normal subgroup of | #72235139, #72238713, #72238720, … |
| maximal quotient of | #72235168, #72287140 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266732);