Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499151 |
| 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 | |
1125 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)(4,10)(5,8)(6,20)(7,12)(9,17)(11,21)(14,28)(15,27)(16,31)(18,23)(19,26)(22,32)(24,30)(25,29);(2,8)(3,14)(4,5)(9,22)(10,12)(11,23)(13,27)(18,19)(20,26)(24,25)(28,31)(30,32);(1,3,5,16,7,14,4,15)(2,9,10,24,12,25,8,22)(6,17,19,32,21,29,18,30)(11,13,26,27,20,28,23,31);(1,4,7,5)(2,10,12,8)(3,15,14,16)(6,18,21,19)(9,24,25,22)(11,26,20,23)(13,27,28,31)(17,30,29,32)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,d^b=di,e^b=ei,f^b=fh,g^c=gi,h^c=hi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72236890, #72237549, #72237551, … |
| maximal normal subgroup of | #72236890, #72237549, #72237551, … |
| maximal quotient of | #72312079, #72337494, C2×#499153 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406349);