Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427434 |
| 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 | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,11,22,10)(3,7,14,12,15,24,16,26)(4,17,19,8,20,18,21,25)(13,27,29,28,30,32,31,23);(1,2)(3,12)(4,18)(5,10)(6,11)(7,14)(8,19)(9,22)(13,28)(15,26)(16,24)(17,20)(21,25)(23,30)(27,29)(31,32);(1,3)(2,7)(4,13)(5,16)(6,15)(8,23)(9,26)(10,12)(11,24)(14,22)(17,27)(18,32)(19,31)(20,30)(21,29)(25,28);(1,4,6,20)(2,8)(3,13,15,30)(5,19,22,21)(7,23)(9,18)(10,17)(11,25)(12,27)(14,29,16,31)(24,28)(26,32)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237065, #72237068, #72237072, … |
| maximal normal subgroup of | #72237065, #72237068, #72237072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334630);