Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427391 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15369 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1269 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)(3,11,14,7)(4,8,17,16)(5,10)(6,9)(12,15,25,20)(13,27)(18,24,21,26)(19,22)(23,30)(28,29)(31,32);(1,2)(3,12)(4,16)(5,19)(6,9)(7,15)(8,17)(10,22)(11,20)(13,28)(14,25)(18,24)(21,26)(23,31)(27,29)(30,32);(1,3)(2,7)(4,13)(5,20)(6,14)(8,23)(9,11)(10,25)(12,19)(15,22)(16,27)(17,30)(18,31)(21,29)(24,28)(26,32);(1,4)(2,8,9,16)(3,13)(5,21)(6,17)(7,23,11,27)(10,26,19,24)(12,28,25,32)(14,30)(15,31)(18,22)(20,29)Polycyclic presentation:
2^9|a^2=e,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338619, #72338625, #72338630, … |
| maximal normal subgroup of | #72338619, #72338625, #72338630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334587);