Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460131 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| 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 | |
1759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(10,13)(11,18)(14,20)(17,19);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20);(1,2,3,4)(5,7)(6,10,12,13)(8,14,16,20)(9,15)(11,17)(18,19);(5,8)(6,11)(7,14,15,20)(9,16)(10,17,13,19)(12,18)Polycyclic presentation:
2^9|c^2=f,d^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72225672, #72225674, #72225680, … |
| maximal normal subgroup of | #72225672, #72225674, #72225680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376879);