Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394470 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,22,20,24)(21,23);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,20)(13,19)(15,22)(16,17)(18,21)(23,24);(9,12)(10,15)(11,18,19,20)(13,21)(14,22,17,24)(16,23);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=f,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72225693, #72225695, #72225697, … |
| maximal normal subgroup of | #72225693, #72225695, #72225697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313923);