Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421074 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15283 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1915 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)(4,15)(5,19)(6,10)(7,13)(8,17)(9,22)(12,29)(14,24)(16,32)(18,27)(20,26)(21,25)(23,31)(28,30);(1,3,5,20,6,13,22,14)(2,7,9,26,10,11,19,24)(4,16,17,30,18,12,21,31)(8,23,15,29,25,28,27,32);(1,4)(2,8)(3,12)(5,21)(6,18)(7,23)(9,27)(10,25)(11,28)(13,16)(14,31)(15,19)(17,22)(20,30)(24,29)(26,32);(3,13)(5,22)(7,11)(9,19)(12,16)(15,27)(17,21)(23,28)Polycyclic presentation:
2^9|b^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72286957, #72287261, #72287274, … |
| maximal normal subgroup of | #72286957, #72287261, #72287274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328270);