Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16893 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12)(3,13)(4,17,20,26)(5,11,7,10)(8,21)(9,18,25,19)(14,22)(15,24)(16,23)(27,29)(28,31)(30,32);(1,3,6,16)(2,8)(4,14)(5,15,7,21)(9,22,25,32)(10,23)(11,13)(12,24)(17,31,26,27)(18,28)(19,29)(20,30);(1,4,5,19,6,20,7,18)(2,9,10,17,12,25,11,26)(3,14,15,29,16,30,21,28)(8,22,23,31,24,32,13,27);(2,10)(4,18)(5,7)(8,23)(9,25)(11,12)(13,24)(14,28)(15,21)(19,20)(22,32)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251784, #72251790, #72251796, … |
| maximal normal subgroup of | #72251784, #72251790, #72251796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381214);