Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498941 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2021 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2131 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)(4,14)(5,18)(8,22)(9,25)(10,15)(12,19)(13,29)(16,28)(20,24)(21,27)(23,32)(26,30);(1,2)(3,8)(4,15)(5,9)(6,10)(7,14)(11,24)(12,23)(13,30)(16,20)(17,25)(18,27)(19,26)(21,31)(22,29)(28,32);(1,3)(2,8)(4,16)(5,19)(6,12)(7,20)(9,26)(10,23)(11,18)(13,14)(15,30)(17,28)(21,32)(22,25)(24,27)(29,31);(3,12)(8,23)(11,19)(13,16)(20,30)(22,32)(24,26)(28,29)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,e^b=ei,h^b=hi,d^c=di,e^c=eh,f^c=fi,g^e=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72236890, #72236924, #72236926, … |
| maximal normal subgroup of | #72236890, #72237529, #72237532, … |
| maximal quotient of | #72312428, #72312697, C2×#498941 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406137);