Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2044 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60753 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1123 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)(3,13)(4,8)(5,19)(6,11)(7,10)(9,16)(12,20)(14,28)(15,29)(17,27)(18,26)(21,23)(22,31)(24,32)(25,30);(2,8)(3,14)(4,6)(5,20)(10,11)(12,26)(15,17)(16,30)(19,23)(22,25)(28,29)(31,32);(1,3,4,15,7,17,6,14)(2,9,11,25,10,24,8,22)(5,16,18,31,20,32,21,30)(12,13,26,28,19,27,23,29);(2,10)(8,11)(9,24)(12,19)(13,27)(22,25)(23,26)(28,29)Polycyclic presentation:
2^9|c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,g^b=gi,h^c=hi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72236900, #72237544, #72237549, … |
| maximal normal subgroup of | #72236900, #72237544, #72237549, … |
| maximal quotient of | #72311884, #72312734, C2×#499126 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406322);