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 | #60426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,13)(4,17)(5,8,6,11)(9,21)(10,20)(14,25)(15,27,30,32)(16,22)(18,23)(19,26)(24,28,31,29);(2,8)(3,14)(4,18)(5,6)(10,17)(11,12)(13,22)(16,21)(19,20)(24,27)(28,29)(31,32);(1,3,7,16)(2,9)(4,15,20,30)(5,21,6,14)(8,22)(10,24)(11,13)(12,25)(17,31)(18,28,19,29)(23,32)(26,27);(1,4)(2,10)(3,15)(5,19)(6,18)(7,20)(8,23)(9,24)(11,26)(12,17)(13,27)(14,28)(16,30)(21,29)(22,32)(25,31)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,f^2=i,g^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 | #72213514, #72216142, #72216181, … |
| maximal normal subgroup of | #72213514, #72216142, #72216181, … |
| maximal quotient of | C2×#385912, #72252103, #72286824 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293108);