Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| socle | C2 | C2×#1790 |
| prime radical | #62759 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,8,14,6,13)(3,15,17,31,18,29,22,30)(4,11,20,27,5,12,19,28)(9,23,25,16,26,32,24,21);(1,3)(2,9)(4,16)(5,21)(6,22)(7,17)(8,18)(10,24)(11,15)(12,29)(13,25)(14,26)(19,23)(20,32)(27,30)(28,31);(2,10)(6,7)(9,24)(11,27)(12,28)(13,14)(15,30)(17,22)(19,20)(23,32)(25,26)(29,31);(1,4,8,5)(2,11,14,12)(3,16,18,21)(6,19,7,20)(9,15,26,29)(10,27,13,28)(17,32,22,23)(24,30,25,31)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fh,g^b=ghi,e^c=ehi,f^c=fi,g^c=gi,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261065, #72261089, #72261203, … |
| maximal normal subgroup of | #72261065, #72261089, #72261203, … |
| maximal quotient of | #72279617 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400821);