Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #385912 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14417 |
| 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 | |
1419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,10,6,11)(3,13,15,24,17,28,16,8)(4,18,20,26,22,27,21,9)(14,30,19,29,32,23,31,25);(1,3)(2,8)(4,19)(5,15)(6,16)(7,17)(9,25)(10,24)(11,28)(12,13)(14,21)(18,30)(20,32)(22,31)(23,27)(26,29);(1,4)(2,9,10,26)(3,14)(5,21)(6,20)(7,22)(8,23,24,30)(11,18,12,27)(13,29,28,25)(15,31)(16,19)(17,32);(2,10)(3,15)(5,6)(8,13)(9,26)(14,31)(16,17)(19,32)(20,21)(23,29)(24,28)(25,30)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216059, #72216130, #72216181, … |
| maximal normal subgroup of | #72216059, #72216130, #72216181, … |
| maximal quotient of | C2×#385914, #72252155, #72252207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293110);