Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394483 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2472 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14727 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13,8,14,6,12)(3,15,19,9,20,26,18,27)(4,21,11,5,22,23,24,10)(16,28,30,32,31,17,25,29);(1,3)(2,9)(4,16)(5,17)(6,19)(7,18)(8,20)(10,28)(11,25)(12,26)(13,15)(14,27)(21,29)(22,31)(23,32)(24,30);(1,4,6,11,8,22,7,24)(2,10,12,21,14,5,13,23)(3,16,18,30,20,31,19,25)(9,17,15,29,27,28,26,32);(1,5)(2,11)(3,17)(4,13)(6,23)(7,21)(8,10)(9,25)(12,22)(14,24)(15,16)(18,29)(19,32)(20,28)(26,31)(27,30)Polycyclic presentation:
2^9|a^2=f,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72263012, #72267112 |
| maximal normal subgroup of | #72263012, #72267112 |
| maximal quotient of | #72262943 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303017);