Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396615 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14735 |
| 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 | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(4,17)(5,7)(9,23)(11,19)(13,21)(14,29)(18,20)(22,25)(24,28)(27,30);(1,2)(3,13,12,9)(4,10)(5,19)(6,11)(7,8)(14,28,27,24)(15,21,16,23)(17,25)(18,26)(20,22)(29,32,30,31);(1,3,5,15,6,12,7,16)(2,9,8,21,11,13,19,23)(4,14,17,30,18,27,20,29)(10,24,22,31,26,28,25,32);(1,4)(2,10)(3,14)(5,17)(6,18)(7,20)(8,22)(9,24)(11,26)(12,27)(13,28)(15,30)(16,29)(19,25)(21,31)(23,32)Polycyclic presentation:
2^9|b^2=f,c^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72338616, #72338625, #72338632 |
| maximal normal subgroup of | #72338616, #72338625, #72338632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303811);