Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #433975 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15494 |
| 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 | |
1035 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)(3,13,17,21)(4,10,20,25)(5,8)(6,11)(7,12)(9,16,24,14)(15,29)(18,26,19,22)(23,30)(27,31)(28,32);(2,8)(3,14)(4,18)(5,6)(10,25)(11,12)(13,21)(16,17)(19,20)(23,29)(27,32)(28,30);(1,3)(2,9,12,24)(4,15)(5,14)(6,16)(7,17)(8,21,11,13)(10,23,25,32)(18,30)(19,28)(20,31)(22,29,26,27);(1,4)(2,10)(3,15)(5,19)(6,18)(7,20)(8,22)(9,23)(11,26)(12,25)(13,27)(14,28)(16,30)(17,31)(21,29)(24,32)Polycyclic presentation:
2^9|a^2=eg,c^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237065, #72237071, #72237086, … |
| maximal normal subgroup of | #72237065, #72237071, #72237086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341171);