Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474328 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16955 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
796 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)(4,8)(5,18,19,9)(6,10)(7,14)(12,20)(13,25,28,21)(15,22)(16,23,29,30)(17,24)(26,32,27,31);(1,3)(2,7,10,11)(4,15)(5,13)(6,14)(8,22,24,20)(9,21,18,25)(12,17)(16,27)(19,28)(23,32,30,31)(26,29);(1,4)(2,8)(3,12,14,15)(5,16,19,29)(6,17)(7,20,11,22)(9,23,18,30)(10,24)(13,27)(21,32)(25,31)(26,28);(1,5)(2,9)(3,13)(4,16,17,29)(6,19)(7,21)(8,23,24,30)(10,18)(11,25)(12,26,15,27)(14,28)(20,31,22,32)Polycyclic presentation:
2^9|a^2=efg,b^2=fgh,c^2=f,d^2=ef,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72318161, #72318171, #72318184, … |
| maximal normal subgroup of | #72318161, #72318171, #72318184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381524);