Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460861 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,15,16,11)(8,12)(9,13)(10,14)(17,19,18,20);(1,3,4,2)(5,7,8,17,10,16,9,18)(6,11,12,20,14,15,13,19);(2,3)(6,12)(7,16)(8,9)(11,19)(13,14)(15,20);(6,13,14,12)(11,19,15,20)Polycyclic presentation:
2^9|a^2=g,b^2=ef,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72287326, #72287341, #72287360, … |
| maximal normal subgroup of | #72287326, #72287341, #72287360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368057);