Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419466 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2021 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,10)(8,11,16,19)(9,12)(13,17,15,20)(14,18);(5,7)(6,10)(8,15,16,13)(9,14)(11,17,19,20)(12,18);(5,8)(6,11,12,19)(7,13)(9,16)(10,17,18,20)(14,15);(2,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,18)(15,16)(19,20)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225628, #72225637, #72225642, … |
| maximal normal subgroup of | #72225628, #72225637, #72225642, … |
| maximal quotient of | #72251457, #72265831, #72265832 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330625);