Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497329 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61059 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#577 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1148 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)(4,16)(5,19,20,8)(6,10)(7,14)(9,17)(12,22)(13,24)(15,26,28,27)(18,25,32,31)(21,29,23,30);(1,3,6,12)(2,7,10,24)(4,13,17,14)(5,15,20,28)(8,23,19,21)(9,22,16,11)(18,29,32,30)(25,26,31,27);(2,8)(3,12)(7,21)(9,25)(10,19)(11,26)(13,14)(15,28)(16,31)(22,27)(23,24)(29,30);(3,13)(4,17)(7,22)(9,16)(11,24)(12,14)(15,29)(18,32)(21,27)(23,26)(25,31)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=ei,g^c=gh,e^d=ei,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72236997, #72237521, #72237542, … |
| maximal normal subgroup of | #72236997, #72237521, #72237542, … |
| maximal quotient of | #72311927, #72312158, C2×#497330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404526);