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:
(2,7)(3,11)(4,15)(5,18)(8,22)(9,16)(10,24)(12,28)(13,30)(14,27)(19,21)(20,23)(25,29)(26,31);(1,2)(3,12)(4,16)(5,19)(6,9)(7,15)(8,14)(10,18)(11,25)(13,20)(17,21)(22,26)(23,27)(24,32)(28,30)(29,31);(1,3,4,13,6,14,15,31)(2,8,7,20,9,12,16,29)(5,11,17,26,18,27,32,30)(10,23,21,22,19,25,24,28);(2,9)(7,16)(8,12)(10,19)(20,29)(21,24)(22,28)(23,25)Polycyclic presentation:
2^9|c^2=e,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, #72237544, #72237553, … |
| maximal normal subgroup of | #72236890, #72237544, #72237553, … |
| maximal quotient of | #72312079, #72312755, C2×#499151 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406347);